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ENGINEERING   & 

MATHEMATICAL 

SCIENCES   LIBR34Ry 


Geometry  and  Coll ine at Ion  Groups 
of  the 
Finite  Projective  Plane, 

by 


■  mL.'  J— 


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Ulysses  Grant  Mitchell. 


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UNIVERSITY  OF  CALIFORNIA 
AT  LOS  ANGELES 


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GEOMETRY 


AND 


COLLINEATION   GROUPS 


OF  THE 


FINITE   PROJECTIVE   PLANE 

PG(2,22) 


A    DISSERTATION 

Presented  to  the  Faculty  of  Princeton  University 

IN  Candidacy  for  the  Degree  of 

Doctor  of  Philosophy 

(department      of      mathematics) 


BY 


ULYSSES  GRANT  MITCHELL 


PRINCETON 
1910 


GEOMETRY 

AND 

COLLINEATION   GROUPS 

OF  THE 

FINITE    PROJECTIVE    PLANE 

PG(2,2^) 


A    DISSERTATION 

Presented  to  the  Faculty  of  Princeton  University 

IN  Candidacy  for  the  Degree  of 

Doctor  of  Philosophy 

(   IJ  E  P  A  R  T  M  E  N  T        OF        M  A  T  H  E  M  A  T  I  C  S    ) 


BY 

ULYSSES  GRANT  MITCHELL 


PRINCETON 
1910 


Copies  of  this  dissertation  may  he  ohtained  on  apphcation  to  the  University 
Ivibrary,  Princeton,  N.  J.  The  price  for  each  copy  is  50  cents,  which  includes 
postage. 


ERRATA 

Page  0,  h'ne  21,  add  -f-d-y^a^^r  to  the  left  member  of  the  equarlon. 

Paire  11.  line     4,   in   place  of   "PG(2,2")"  read  "PG(2,2  )." 

Page  12.  line    10.   in   place  of   "p.v..'^A\,-(— *•:/    read  ''px./=ix.,-\-x..'' 

Page  12,  line    11,   in  place  of    "((0-'-|-/3-f~i )  '   read   "  (p '-f-^j-f /)."           « 

Page  18,  line  30,   in   place  of   "on"-  read  "or."                                               ^ 

Page  20.  line   2^,   in   place  of   "/„"  read  "/."    (twice).                                ^ 


it* 


Press  or  The  Journal-Wori  d 

Lawrence,  Kansas 

1914 


Engineering  & 
Maihemali**! 

Sciences 

Library 


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189455 


GEOMETRY  AND  COLLINEATION  GROUPS  OF  THE  FINITE 
PROJECTIVE  PLANE  PG  (2,2-).* 


§1.  Definition  of  a  Finite  Projective  Plane. 

§2.  Preliminary  Theorems. 

§3.  Types  of  Collineations  in  PG  (2,2"). 

^4.  Cyclic  Groups  in  PG  (2,2-), 

§5.  The  Group  of  Determinant  Unit}- — Gooieo- 

^6.  The  Group  Leaving  Invariant  an  Imaginary  Triangle — Ggg. 

1^7.  Invariant  Real  Configurations  and  Their  Groups. 

^S.  Subgroups  of  the  Group  Go^go  Which  Leaves  a  Line  Invariant. 


§/.  Definition  of  a  Finite  Projective  Plane. 

The  definition  and  general  properties  of  finite  projective  spaces  together  with 
references  to  the  literature  of  the  subject  may  be  found  in  a  paper  by  veblen 
and  BUSSEY  in  the  Transactions  of  the  American  Mathematical  Society,  Vol.  7, 
pp.  241-259.  They  used  the  symbol  PG(l<;,p"),  where  k,p,n  are  integers  and  p  is 
a  prime,  to  indicate  a  finite  projective  space  of  k  dimensions  having  p"-|-  1  points 
to  the  line.  It  is  the  purpose  of  this  paper  to  discuss  some  of  the  properties  of  the 
PG(2,2")  and  to  determine  all  subgroups  of  the  group  of  projective  collineations 
in  PG(2,2^). 

We  give  a  brief  summary  of  the  analytic  and  synthetic  definitions  of  a  finite 
projective  plane. 

If  x^.Xn.x..,  are  marks  of  a  Galois  fieldt  [designated  by  GF(p")]  of 
order  p"  there  are  (p''" — l)/(p — 1)=P""  +  P"+1  elements  of  the  form  (x^jX^^x^) 
provided  that  the  elements   {x^,x.^,x^)    and   {h\Jx^,lx^,)    indicate  the  same  element 


*  Presented  to  the  American  Mathematical  Society,  April  29,  1911 

•^  For  definition  and  properties  of  a  Galois  field  see  E.  H.  MOORE,  Subgroups  of  the 
Generalized  Finite  Modular  Group,  University  of  Chicago  Dec.  Pub.  Vol,  IX.,  pp.  141- 
156;  L.  E.  DICKSON,  Linear  Groups,  pp.  1-14. 


2  U.  G.  Mitchell:  Geometry  axd 

when  /  is  any  mark  other  than  zero  and  provided  that  {0,0, u)  be  excluded  from 
consideration.     These  elements  constitute  a  finite  projective  plane  if  the  equation 

u  ^X^-\-U.,X.,-{-U^X.^=() 

[the  domain  for  coefficients  and  variables  being  the  GF(p")]  be  taken  as  the  equa- 
tion of  a  line  except  when  u^  =  u.,^u.,=^o.  The  line  is  denoted  by  the  symbol 
(«,.?/.,.«;()  and  the  symbols  {u^,u.,,ii..)  and  {hi^Ju.^u ..)  where  /  13  any  mark  other 
than  zero  denote  thr  same  line.  The  points  of  a  line  are  those  points  whose  co- 
ordinates (.Vj.Ao.A-^)  satisfy  its  equation. 

Taking  0,1,  i  and  /-  for  the  marks  of  the  GF  (2-)  where  7  is  defined  as  a  root 
of  the  equation  /-=/-l-  /  and  hence  r^=/  (mod.  2)  the  PG(2,2-)  so  defined  may 
be  exhibited  in  the  table  of  alignment  given  on  the  opposite  page.  In  the  analytical 
processes  of  PG(2,2")  no  distinction  need  be  made  between  plus  and  minus  signs 
since  -1^1    (mod.   2). 

Synthetically  a  finite  projective  plane  may  be  defined  as  a  set  of  elements  which 
for  suggestiveness  are  called  points,  arranged  in  subsets  called  lines  and  subject  to 
the  following  conditions: 

I.  The  set  contains  a  finite  number,  greater  than  one,  of  lines,  and  each  line 
contains  p"-(-l  points  (p  and  n  integers  and  p  a  prime). 

II.  If  A  and  B  are  distinct  points  there  is  one  and  only  one  line  that  contains 
A  and  B. 

III.  All  tlie  points  considered  arc  in  the  same  plane. 

From  this  definition  it  follows*  that  the  principle  of  duality  i:'. 
^■alid  in  the  plane  so  defined,  that  there  are  p"-|-l  lines  through  each  point  and 
that  the  total  number  of  points  in  the  plane  is  p'"'"-|-p"-i-l. 

In  a  PG  (2,2-)  there  are  then  21  points  and  21  lines.  The  following  set  of  ele- 
ments arranged  in  21  lines  of  o  elements  each  will  be  seen  to  satisfy  the  given 
synthetic  definition  and  to  be  identical  with  the  table  given  opposite. 

0  1     2     ;{  4     ;")  (i     7     S     9     1(1  11  12  13  14  lo  16  17  18  19  20 

1  2     3     4  5     0  7     8     1)     10  11  12  13  14  15  16  17  18  19  20  0 
4     5     6     7  <S     9  10  11  12  13  14  15   Ki  17  IS  19  20  C     1     2  3 
14  15  16  17  18  1!)  2(1  0     12     3     4     5  6     7     8     9     10  11  12'  13 
16  17  18  19  20  0  12     3     4     5     (i     7  8     9     10  11  12  13  14  15 

^2.  Preliminary   Theorems. 

Theorem  1.  //;  FG(2,2")  the  diagonal  points  of  a  complete  quadrangle 
are  collinear. 

Proof.  Let  three  of  the  vertices.  A,  B  and  C  of  the  quadrangle  (Fig.  1)  be 
taken  as  the  triangle  of  reference  and  let  the  fourth  vertex,  D,  be  assigned  the  co- 
ordinates {1,1,1).    The  intersections  of  AB  with  CD,  AC  with  BD  and  AD  with 


Cf.  VEBLEN  and  YOUNG,  Projective  Geometry,  Vol.  I,  pp.  16-17. 


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4  U.  G.  Mitchell:  Geometry  and 

BC      determine      the      diagonal      points      P,Q,R      as       (/,0,/),       {i,i,o)       and 
(o,/,/)    respectively,  which  are  collinear  on  the  line  a-,-}-a"2+-^"3=^'5- 


B(ooi) 


FlG.l. 

Since  the  quadrangle  A,B,C,D  is  projectively  equivalent  to  any  other  quad- 
rangle in  the  plane  the  proof  is  complete. 

The  line  joining  the  diagonal  points  of  any  complete  quadrangle  will  be  re- 
ferred to  as  the  d'lagonol  line  of  the  quadrangle. 

Definition  of  conic.  A  point  conic  is  defined  as  the  locus  of  the  points  of  in- 
tersection of  corresponding  lines  in  two  projective  non-perspective  pencils  of 
lines.  A  line  conic  is  defined  as  consisting  of  the  lines  that  join  corresponding 
points  in  two  projective  non  perspective  ranges  of  points.  In  PG  (2,2")  the 
number  of  points  in  a  point  conic  is  2"-)-l,  the  number  of  lines  in  a  pencil,  and 
the  number  of  lines  in  a  line  conic  is  2"-|-l .  the  number  of  points  in  a  range. 
Hence  in  PG(2,2-)  a  point  conic  consists  of  any  five  points  no  three  of  which 
are  collinear  and  a  line  conic  consists  of  any  five  lines  no  three  of  which  are  con- 
current. 

A  tangent  to  a  point  conic  is  defined  as  a  line  which  has  one  and  only  one 
point  in  common  with  the  conic.  There  is  one  and  only  one  tangent  at  a  given 
point  on  a  conic  since  there  are  2"-|-l  lines  through  the  point  and  2"  lines  join- 
ing it  to  other  points  of  the  conic. 

By  taking  the  equations  of  two  projective  pencils  of  lines  \P-\-fx0^o  and 
\P'-\-p.Q'=0  [P=0,  Q=o.  P'=o,  Q'=0  being  equations  in  abbreviated  no- 
tation oi  lincj  in  PG(2,1.^'')  and  A  and  /x  marks  of  the  GF(2")]  and  eliminating 
A  and  /i  it  is  readily  shown  that  the  equation  of  a  point  conic  is  a  homogen- 
eous equation  of  the  second  degree  in  three  variables  with  coefficients  in  the 
GF(2").     Similarly,   using  line  coordinates  it    may  be  shown    that   the   equation 


COLLINEATTON    GrOUPS   OF    PG(2,2-).  5 

of  a  line  conic  is  also  a  homogeneous  equation  of  the  second  degree  in  three  vari- 
ables with  coefficients  in  the  GI'(2"). 

Theorem    2.    Every  equation  of  the  form 

1,2,3, 

i.j 
where  the  coefficients  are  marks  of  the  GF(.2^)   is  satisfied  by  the  coordinates  of 
at  least  one  point  in  PG(2,2^). 

Proof.  Suppose  neither  ^i^  nor  «,,  to  be  zero.  Taking  x,^=i  the  equation 
reduces    to    «iiXi--|- («j2-*'2~l~^i3)-^i~l~'^22'*'2""]"'^23*'2~f''^33^^^    which    is    satisfied    by 

*"'""    «ii«i2     y  (' .i'<^h^(^:'--\-(ii2(ii^a2^-\-(ii2'(^zz) 
and     x^=  -^        Moreover,     since     in    the     GF(2")      every      mark     satisfies      the 

n 

equation  x-  ^sx  it  is  a  perfect  square  and  since  — i^i  (mod.  2)  its  square  root 
is  unique.  These  values  for  at^  and  x.y  are  therefore  uniquely  determined  and 
lie  in  the  GF(2").       If   a^^^=0,   F{x.^,x.,,x.^)=0   is   satisfied    by    {i,0,o)    and    if 

/^  ^ 

«ii  ±  0  and  a^„=0  h\x ^,x^,x.,)=o  is  satisfied  by  (  \  7~2    ^^'^i' 

"n 
Theorem     3.    Every  equation  of  the  form 

1,2,3, 

F(x^,x.,,xJ^     S     a\iX^x  ^=0  (i-gi) 

where  Xy,Xn,x^  are  point  coordinates  and  the  coefficients  are  marks  of  the  GF(2^) 
represents  a  point  conic  in  PG(2,2^). 

Proof.  Since  by  the  previous  theorem  F{x^x^,x^)=0  is  satisfied  by  at  least 
one  point  in  PG(2,2"),  by  means  of  a  linear  transformation  of  that  point  into  the 
point    (o,o,/)   F{x.^,Xr,,x^)=o  can  be  transformed  into  the  equation 

F^{x^,Xn,Xr^)    =    b^^x^--\-b^nX^Xo  '^b.,nX^~-\'b-^.^x.^x^-\-b^^x.-.x^=o 
This  equation  may  be  written 

x,{b,,x,^b,^_x^_^b^^x^)    +   x^_{b  22Xi-\-b2zX.,)=0 
which  is  seen  to  be  the  locus  of  points  of  intersection  of  corresponding  lines  in  the 
two  projective  pencils  of  lines 

'-^2+'«  (^ii^i+^i2-*'2+^i3*"3)   "=0  and  lx^-{-m  (^22^2+^23*3)=^ 

Hence  F{x^,Xn.,x^)=o  represents  the  locus  previously  defined  as  a  point  conic. 

Theorem    4.    In  PG(2j2")  all  tangents  to  the  point  conic 

1,2,3, 
F(x^,Xo,xJ^^     S     aij;ViArj=o  d^^) 

i.j 

are  concurrent   and  their  point   of  concurrence   is    (ci^^'^ii'^  vj) • 


6  L .  G.  Mitchell:  Geometry  axd 

Pi  oof.  The  line  <"i^i+^2'*"2~l~^3-^3='''  ^^^  domain  for  variables  and  coefficients 
being  the  GF(2").  will  be  a  tangent  to  the  conic  Y  {x^,x^,x^)^=o  in  case  it  has  but 
one  point  in  common  with  the  conic.     Eliminating  x^  from  the  two  equations  gives 

(«11^2'    +    «12^1^2)     *2"    +    ^1     («12^3    +    «l.-i<"2    +    '^23''l)       *2*3      + 

The  condition  that  this  shall  be  a  perfect  square  is  (assuming  r,  not  o) 
^i"2z  +  ^^13^2  H~  ''3'^i2=^  which  is  seen  to  be  the  condition  that  the  line 
i\x^  -\-  foATo  4"  '^z'^i^^o  pass  through  the  point  («23'''i3'^i2)-  ^^  '^\^=0  either 
fo  or  fg  is  not  zero  and  ,Vo  or  a%  can  be  eliminated. 

If  an  outside  point  of  a  conic  be  defined  as  any  point  of  intersection  of  tangents 
lu  the  conic,  it  follows  that  in  PG(2.2")  a  conic  has  but  one  outside  point.  Hence 
the  point  of  concurrence  of  tangents  to  a  conic  will  be  referred  to  as  the  outside 
point  of  the  conic. 

Corollary  i.  Every  line  through  the  outside  point  of  a  conic  is  a  tangent  to  the 
conic. 

Corollary  2.  Through  any  point  in  PG  (2,2^)  other  than  the  outside  point  of 
a  given  conic  passes  one  and  but  one  tangent  to  the  conic. 

Corollary  J.  In  PG(2,2")  the  condition  for  degeneracy  of  a  conic  is  that  the  co- 
ordinates of  its  outside  point  satisfy  its  equation. 

For  the  general  conic  F(Ar,,Av.Ar.^)=--o  the  condition  is 

'Ml'^23"+'^12'^23'^l.-'.~l~^;;.i'^12""=0 

Corollary  4.  In  PG(2,2")  six  and  but  six  points  can  be  chosen  such  that  no 
three  of  the  set  are  coUinear. 

For.  any  five  points  no  three  of  which  are  collinear  determine  a  conic  which  to- 
gether with  its  outside  point  constitutes  a  set  of  six  points  no  three  of  which  are 
collinear.  The  existence  of  any  other  point  not  collinear  with  any  two  of  these 
contradicts  Corollary  -  above. 

Corollary  5.  In  PG(2.2-)  the  diagonal  line  of  the  complete  quadrangle  of  any 
four  points  of  a  conic  is  the  tangent  of  the  fifth  point. 

For,  in  PG(2,2-)  there  are  five  points  to  the  line  and  hence  the  diagonal  line 
contains  two  points  other  than  the  diagonal  points.  These  two  points  together 
with  the  four  points  determining  the  quadrangle  form  a  set  of  six  points  no  three 
of  which  are  collinear.  It  follows,  then,  from  Corollary  4  that  the  two  points  on  the 
diagonal  line  arc  the  fifth  point  and  the  outside  point  respectively  of  any  conic 
passing  through  the  points  of  the  quadrangle. 

§  3-  Types  of  CoUiniations  in  PG(2,2^). 
Transformations  on  the  line.     To  determine  the  one-dimensional  transformations 
in   PG(2.2")   it  is  sufficient  to  consider  the  transformations  of  points  on  the  line 
x^=o  and  any  point  on  it  can  be  represented  by  two  coordinates.     Hence  the  gen- 
eral projective  transformation  of  points  on  a  line  in  PG(2,2")  may  be  written 

T:  p.Vi'=rt.^Ar,-|-rti.,Aro,  (i=i,2), 

where    the    determinant    A^=rtii      of    the    transformation    is    not    zero    and    the 


COLLINEATION    GrOUPS   OF    PG(2,2-).  7 

domain  for  variables  and  coefficients  is  the  GF(2").  In  the  usual  man- 
ner we  put  Xi'=-Xi.  (/=1,2)  in  order  to  determine  the  fixed  elements  and  con- 
sider the  characteristic  equation 

P^+(«11+«2.)P+A=0  (1) 

which  expresses  the  condition  for  consistency. 

According  as  (1)  has  one,  two,  or  no  roots  in  the  GF(2")  T  has  one,  two,  or  no 
fixed  points  in  PG(2,2")  and  is  designated  correspondingly  as  parabolic,  hyper- 
bolicov elliptic.  lnPG(2,2")thereare  2"  (2"-l)  equations  of  the  form  (1)  since  A  is 
not  zero  and  there  are  2"-l  marks  other  than  zero  in  the  GF(2"').  One  half  of  these 
equations  have  both  roots  in  the  GF(2")  and  the  other  half  have  no  roots  in  the 
GF(-2").  Of  those  having  roots  in  the  GF(2"),  2"-l  have  coincident  and 
(2"-l)  (2""'-l)  have  distinct  roots.  The  necessary  and  sufficient  condition  that 
(1)  shall  have  coincident  roots  is  that  rtii=^rtoo(mod.2).  When  rtn=^22  is  substi- 
tuted in  T  it  is  found  that  T"  becomes  the  identical  collineation  and  hence  every 
parabolic  transformation  in  PG(2.2")  is  of  period  two.  An  hyperbolic  transforma- 
tion permutes  2"— 1  points  and  hence  its  period  is  2"— 1  or  some  factor  of  2"-!. 
Similarly  the  period  of  an  elliptic  transformation  must  be  2"-)-l  or  some  factor 
of  2"-f  1. 

Suppose  (1)  to  be  irreducible  in  the  GF(2").  From  the  theory  of  Galois  fields 
we  know  that  its  roots  then  are  marks  p^,  /aj-"  of  the  GF(2-")  conjugate  with 
respect   to   the   GF(2").      Substituting   these   values   in   T   we   find   the   invariant 

points  of  T  to  be  («io,flj,+/j, )  and  {a^.,,a^^Arpy'") ■  There  are,  therefore,  on  the 
line  2"(2"-l)  points  of  PG(2,2-")  (referred  to  as  "imaginary"'  points)  arranged 
in  2"-^  (-"-!)  pairs  of  conjugate  points  which  figure  as  the  double  points  of  the 
elliptic  transformations.     Hence  in  PG(2,22)  there  are  six  such  pairs  on  each  line. 

As  in  ordinary  projective  geometry  it  can  be  proved  that  any  three  points  on  the 
line  in  PG(2,2")  can  be  transformed  into  any  other  three  points  of  the  line  and 
that  if  three  points  of  the  line  are  fixed  all  points  of  the  linf"  are  fixed.  From  this 
it  follows  that  the  number  of  parabolic  transformations  is  2""-  1,  of  hyperbolic 
transformations  is  2"-i  (2"-2)  (2"-|-l)  and  of  elliptic  transformations  is  2-"(2"-l)/2. 
The  total  number  of  callineations  on  the  line  is  the  total  number  of  distinct  trans- 
formations T  having  coefficients  in  the  GF(2")  and  determinant  not  zero.  This 
is  determined  to  be  2"(2"'''-]). 

In  PG(2,2-)  according  to  the  above  there  ore  GO  transformations  on  the  line 
ar.d  of  these  15  are  parabolic,  20  hyperbolic  and  24  elliptic.  Since  the  total 
t-roup  is  of  order  00  and  can  be  exhibited  as  permutations  of  the  five  points  of 
the  line,  it  must  be  the  alternating  group  on  five  symbols  and  hence  its  subgroups 
are  well  known.  They  will,  however,  be  enumerated  later  in  determining  the 
groups  which  leave  a  line  invariant. 

Transformations  in  the  plane.  The  general  linear  homogenous  transformation 
in  PG(2,2")  may  be  written 


8  U.  G.  Mitchell:  Geometry  and 

where  the  determinant  A=|rtij|  of  the  transformation  is  not  zero  and  the 
domain  for  the  coefficients  is  the  GF(2").  To  determine  the  invariant 
points    we    set    A'i=.Vi,    (i=l,2,3)    and    obtain    the    characteristic    cubic 

p^   +    ia,,+a,,^a,,)    p^    +    iJ^^J^J.^^_^J^^)    p+A=0  (2) 

where  J^  is  the  co-fac*^or  of  (7ij  in  A^=|rtijl.  Since  there  are  2"  marks  in  the 
GF(2")  there  are  2-"(2"-l)  equations  of  the  form  (2)  belonging  to  the 
GF(2").  Of  these  2"-l  have  all  three  roots  coincident,  (2"-l)  (2"-2)  two  roots 
coincident  and  the  other  distinct,  (2" — 1)(2" — 2(2" — 3)/3;  all  three  roots  dis- 
tinct, and  2"(2"-l)-/2  one  root  in  the  GF(2")  and  two  roots  in  the  GF(2-")  but 
conjugate  with  respect  to  the  GF(2").  These  last  are  made  up  of  the  products 
of  the  2" — 1  linear  factors  in  GF(2")  with  the  2"(2"  i)/2  irreducible  quadratics 
which  appeared  as  characterictic  equations  of  transformations  on  the  line.  There 
are  then  2"(2"— 1)(2"-'— l)/;}cubics  (2)  having  roots  in  GF(2")  or  GF(2-"). 
The  remaining  2"(2-"-l)/o  are  irreducible  in  the  GF(2")  but  from  the  theory  of 
Galois  fields*  it  follows  that  their  roots  are  marks  of  the  GF(2'")  conjugate  with 
respect  to  the  GF(2").    If,  therefore,  A  be  a  root  of  an  irreducible  cubic  belonging 

to  the  GF(2")  its  other  roots  are  A""  and  A""",  where  A  is  a  mark  of  the  GF(2''"). 
If  we  put  Xi'=Xi,  (/=1,2,3)  in  T,  and  substitute  A  for  p  we  obtain 

as  the  corresponding  invariant  point.  The  other  invariant  points  are  then  neces- 
sarily the  points  obtained  by  substituting  for  A  in  this  expression  A""  and  A'  "  re- 
spectively. Hence  every  transformation  T,  whose  characteristic  equation  (2)  is 
irreducible  in  the  GF  (2")  leaves  invariant  a  triangle  in  PG(2,2'">)  which  will  be 
designated  as  an  imaginary  triangle  to  indicate  that  it  is  not  in  PG(2,2"). 

Corresponding  to  the  three  cases  in  which  the  characteristic  equation  (2)  has 
three  distinct  roots  there  are  then  three  types  of  transform.itions  having  for  in- 
variant figure  a  triangle.  These  will  be  designated  as  type  I„,  type  I,,  type  I3  ac- 
cording as  the  invariant  triangle  has  none,  one,  or  three  of  its  vertices  in  PG(2,2"). 

If  the  equation  (2)  has  two  roots  coincident  the  corresponding  coUineation  is 
designated  type  II  and  leaves  invariant  two  points  and  by  duality  two  lines.  Its 
invariant  figure  has  two  points  on  one  line  and  two  lines  on  one  point.  If  the 
three  roots  of  (2)  coincide  the  corresponding  transformation  leaves  invariant  a 
lineal  clement  and  is  called  type  III.  Two  special  cases  of  the«c  will  be  classified 
as  separate  types  because  of  their  importance.  A  transformation  other  than  the 
identity  which  leave«  invariant  all  points  of  a  line  /  called  the  axis  and  all  lines 
through  a  point  P  called  the  center,  is  called  a  homology  or  type  IV.  Such  trans- 
formations appear  among  the  powers  of  those  of  types  I,  and  II.  A  transforma- 
tion, other  than  identity,  which  leaves  invariant  all  points  of  a  line  /  and  all  lines 
through  a  point  P  on  /  is  called  an  elation  or  type  V.     The  point  P  and  the  line 


*  Cf.  Dickson,  1.  c,  p    21  and  p.  53. 


COLLINEATION    GrOUPS   OF    PG(2,2-).  g 

/  are  called  the  center  and  axis,  respectively,  of  the  elation.    Such  transformations 
appear  among  the  powers  of  those  of  types  II  and  III. 

The  invariant  figures  of  the  different  types  are  shown  in  Fig.  2.  Fixed  lines 
which  are  imaginary  are  dotted  and  fixed  points  which  are  imaginary  are  left 
open. 


X 


n 


The  following  formulae*   for   the  number  of   transformations  of   each   type  in 
PG(2,2")  are  readily  obtained: 


*  Cf.  Dickson,  1.  c,  pp.  237-239, 


10  U.  G.  MircHEiL:  Geometry  and 

Nl,=2*"(2-^"— 1)    (2»— 1)   /2 

Ni3=23n(2=»4-2"+l)    (2"+l)    (2'>— 2)    (2"— 3)/6 

Nii=2=="(23"— 1)    (2"-fl)    (2"— 2) 

Niii=2"(23"— 1)    (2="— 1) 

Niv=2-"(2-"+2"+l)    (2"— 2) 

Nv=(2'"— 1)    (2"+l) 

Identlty=l 

Total=2^"  ( 2  *"— 1 )     ( 2-"~l )  ^ 

According  to  these  formulae  the  order  of  the  total  group  of  PG(2,2-)  is  60480 
distributed  as  follows: 

Ni„=in200 

Nil =24192 

Ni,=  2240 

Nil=lCOSO 

Nni=3780 

Niv=    672 

Nv=      315 

Identity=l 
Total=604S0 
The   group   of   all   projective   transformations   in    PG(2,2-)    will   be   designated 
as  vjTgQ^gQ. 

§4.  Cyclic  Groups  in  PG(2,2-). 

We  wish  to  determine  in  detail  the  path-curves  and  periodicity  of  each  of  the 
l\pes  in  PG(2,2-).  In  so  doing  we  shall  at  the  ^~ame  time  determine  all  of  the 
cyclic  subgroups. 

Type  I^j     Consider  the  coUineation 

pA-..'=;-'A-,  -\-i-x.-,-\-x.. 

The  determinant  of  T,,  is  A=i-.  Its  characteristic  equation  p^'-\-ip--\-i'p-\- 
i'-=0  is  irreducible  in  the  GF(2-)  and  has  for  roots  v*'' ,v''^ yV*"'-  where  ^'  is  a 
primitive  root  of  the  GF(2*')*  and  hence  v-^=i,  v^^=^.  The  invariant  point? 
cf  To  are,  therefore,  A,=  {l,v\v'),  B„^-:(l,z;"V-^)  and  C,  =  (  ^.7■'^■^■•'").  T„  is  of 
period  21  and  permutes  the  points  or  PG(2,2-)  in  the  order  of  their  numbering 
in  the  table  of  alignment.  There  is,  accordingly,  some  power  of  T„  which  will 
transform  any  given  point  of  PG(2,2-)  into  any  other  given  point  of  PG(2,2-). 
To  show   that   every  coUineation   of   type    I,,   in   PG(2,2-)    is   conjugate   to   some 

*  For  Galois    field    tables,    see   an   article  hy  VV.  H.  BussEV,  in  the  Bit//.  Amer.  Math.  Soc, 
Vol.  XI,  p.  27. 


COLLINEATICIN    GrOUPS    OF    PG(2,2-).  11 

powerof  T,j  it  is  only  necessary  to  show  that  an\  triangle  A^(  A  J, Ao, A;.),  B;^(  Ai*,A2*,Ao/), 
C^(Ai^",A2^",  AJ")  where  ApA.,A.  are  any  three  marks  in  the  GF(2")  linearly 
independent  with  respect  to  the  GF(2-'),  can  be  transformed  into  A,,,Bo,C„,  re- 
spectively, by  a  transformation  in  PG(2,2").  The  condition  that  A,, A:... A.,  be  lin- 
early independent  with  respect  to  the  GF(2-)  is  necessary  because  it  will  be  ob- 
served that  if  the  coordinates  of  the  point  A  satisfied  the  equation  a^x^-]-a.,x^-{- 
^■_iX.^^^O  those  of  B  and  C  would  also  satisfy  the  equation 

2  4 

[a^x^-{-a.A•.-\-a..x..)-  ^  (aiX^-\-a.,x.,-\-a.,x,.)-  ^^((i^Xi-{-fi.^x..,-\-a.,x.^)^0, 

and  any  transformation  leaving  A,  B  and  C  invariant  would  leave  invariant  a 
line  of  PG(2,2-)   and  therefore  not  be  of  type  I,,. 

i=3 
The  conditions  that  T=/3A-i'^XrtijA-j.      (/=12,3),  ](?(,[    not  0,  where  every  a^ 

i=i 
is  in  the  GF(2-')   shall  transform  A  into  Ay  are 

<',';. iA,+^/..X+rt3;,A^=?''    (a,^^,^a,.u\.^a,,,\.,)  { ^^^ 

rt  , ,  A,  +^/:.,oA2+^33'^3='^'''''  (  ^'-21^^  +^'iii:'^2  +  '''-:A:i  )  j  ' 

which  raised  to  the  2-  and  2*  powers,  are  seen  to  be  the  conditions  that  T  trans- 
form B  to  B„  and  C  and  C,,.  In  (4)  we  may  assign  «.,,,  «.,._,  and  a^^  arbitrarily 
In  the  GF(2-)  provided  not  all  are  taken  as  zero.  We  then  have  ^HiAi+rtaoAo-f- 
^/ggA3=z'''  some  mark  other  tlian  zero  of  the  GF(2'').  Since  Ai,Ao,A3  are  three 
marks  of  the  GF(2'')  linearly  independent  with  respect  to  the  GF(2-)  it  is  pos- 
sible to  choose  rtu,"?).,  and  n^..  v/ithin  the  GF(2-)  such  that  (in'^i~\~^'v2^-2~\~^'v,iK 
is  any  mark  of  the  GF(2".)*  Accordingly  we  take  c^,,.c/|o  and  «!.,  such  that 
'^ii'^i4-  ^i2'^2+^i3'^3^'^''  '  '"""^  similarly  a.,^,  a.,,  and  a.,.^  such  that  a^^X^A;;-a.^.X.,-\-ao.:^\.,_ 
=<r;'^-«\  The  desired  transformation  T  is  thereby  determined  within  the 
GF(2-).     Moreover  the  determinant  of  T  is  not  zero  since 

^31  A, +rt,5oAo-|-«3:iA3^'^''' 

form  a  set  of  simultaneous  non-homogeneous  equations  in  A,, A,  and  A.. 

The  21  powers  of  T„  form  a  cyclic  subgroup  of  Geo48c  and  since  the  triangle 
ABC  can  be  chosen  in  2" ( 2*-l )  (2--l)/3  different  ways  there  are  OliO  such  conju- 
gate cyclic  subgroups  in  G(;„48o- 

Since  the  determinant  of  T„  is  r  the  determinant  of  T„-  is  2^==/  and  the  de- 
terminant of  T,,'*  is  1.  The  powers  of  Tp,  then,  which  are  also  powers  of 
Tq^  and  no  others  are  of  determinant  unity.  The  group  Go,(cyc.  IJ  consisting  of 
the  21  powers  of  T„  contains  accordingly  a  self -conjugate  subgroup  of  order  7 
consisting  of  the  7  powers  of  T„^     G.j,,^,,,  must  contain  91)0  such  cyclic  subgroups. 

Again,  To^  is  of  period  3.      Hence,  Go,    (eye.  I^,)    contains  a  cyclic  subgroup  of 
order  3  which  must  also  be  self-cou jugate  since  no  others  powers  of  T„  than  pow- 
ers of  T„"  are  of  period  3.     G„„^s(i  "ii^ist  contain  9G0  such  conjugate  subgroups. 
*  Cf.  Dickson,  I.  c,  p.  49 


12  U.  G.  Mitchell:  Geometry  and 

To^  must  permute  all  points  of  PG(2,2-)  in  triangles  since  if  it  permuted  any 
three  collinear  points  among  themselves  it  would  leave  invariant  the  line  joining 
them.  It  will  be  seen  later  (in  discussing  the  simple  group  G^q^)  that  a  trans- 
formation of  type  Iq  of  period  7  permutes  among  themselves  seven  points  so  re- 
lated that  for  every  four  of  the  points  which  are  no  three  collinear  the  other  three 
are  the  diagonal  points  of  their  complete  quadrangle. 

Type  I^.     The  coUineation 

T,  :      px./=x^ 

px./=x.,-\-x.^ 
has  the  characteristic  equation  (/o+l)  {p'-\-p-\-^)^0  whose  roots  are  1,  u  and 
u^  where  u  is  a  primitive  root  in  the  GF(2*)  and  hence  u^^i  and  u^-';^i.  The 
invariant  points  of  T,  are  A^^  (7,0,0),  B,s^  {o,i,u),Ci^  (0, /,«*),  and  T,  is 
therefore  of  type  I^  with  A,  for  center  (or  invariant  real  point)  and  Ari=0  for 
axis  ,or  invariant  real  line).  Tj  is  of  period  15  and  T,^  and  Ti'"  are  homolo- 
gies. Accordingly,  the  group  G^.^  (eye.  I^)  consisting  of  the  15  powers  of  T^ 
contains  a  self -con  jugate  cyclic  subgroup  of  order  3  containing  two  homologies 
and  the  identity.  Since  the  determinant  of  Tj  is  ij  T^^,T^^,'1\°,T^^-,  T^^^  and  no 
other  powers  have  determinant  unity  and  are  of  period  5  with  the  exception  of 
T^^^^I.  Gi5(cyc.  Ij)  therefore  contains  a  self-conjugate  cyclic  subgroup  of 
order  5  consisting  of  these  transformations. 

Tj^,  which  is  of  period  5,  permutes  the  lines  through  A,  in  cyclic  order  and 
hence  a  point  Pj  not  on  the  axis  has  4  other  conjugates  P2,P3,P4,P,i  such  that  no 
two  of  the  paints  Pj,P2,P3,P4,P-,  are  collinear  with  A,.  Moreover,  no  three  of 
the  points  P,,Po,P3,P4,P,r,,  can  be  collinear,  for  if  they  were  the  line  containing 
them  would  be  invariant  under  T^''.  Hence,  Pi,Po,PH,P4,P,r,  form  a  point  conic 
having  Aj  for  outside  point.  Evidently  T^^  leaves  invariant  three  such  point 
tonics  having  A^  for  outside  point  and  by  duality  three  line  conies  having  Xi^o 
for  outside  line. 

Every  coUineation  T/  of  type  T,  in  PG(2,2-)  is  conjugate  to  Tj  or  some 
one  of  its  powers  since  there  is  in  PG^2,2-)  a  transformation  S  transforming  any 
point  P'  and  line  /'  into  {1,0,0)  and  x^=o  respectively  and  a  transformation  S,, 
leaving  the  point  {1,0,0)  fixed  and  changing  any  pair  of  conjugate  imaginary 
points  on  a:j=o  into  the  pair  {o.i,u*).  The  coUineation  (SS,)T/(SS,)  '  must 
then  be  some  power  of  T^. 

In  discussing  the  one-dimensional  transformations  it  was  shown  that  in 
PG(2,2-)  there  are  six  pairs  of  conjugate  imaginary  points  on  each  line.  Hence 
there  are  In  PG(2,2-)  21 46 -G  =201  (i  conjugate  groups  Gj-(cyc.I,)  each  con- 
taining a  cyclic  self-conjugate  subgroup  of  order  5  consisting  of  the  transformations 
of  period  5,  and  a  cyclic  self-conjugate  subgroup  of  order  -T  consisting  of  the  hom- 
ologies. There  are  2016  of  the  cyclic  subgroups  of  order  5  but  only  21  -16=336  of 
the  subgroups  of  order  3  since  the  same  subgroup  of  order  3  appears  with  every 
Gj5(cyc.I,)   which  leaves  invariant  a  given  center  and  axis. 


COLLINEATION    GrOUPS   OF    PG(2,2-).  13 

Type  I3.  If  T3  be  a  transformation  of  t^^pe  I,  it  leaves  invariant  a  real  trian- 
gle A,B,C  T3  is  fully  determined  by  its  Invariant  triangle  and  the  transforma- 
tion of  a  point  P  into  a  point  P'  provided  the  points  A,B»C,P  andP'  are  no  three 
col  linear.  Hence,  (Theorem  4,  Cor.  4)  there  are  two  choices  for  P'  for  a 
given  point  P.  Accordingly  T,,  is  of  period  three  and  permutes  the  9  points  not 
on  the  sides  of  its  invariant  triangle  in  three  triangles.  It  should  be  noted  that 
any  one  of  these  triangles  together  with  the  points  A,B,C  form  a  set  of  six  points 
no  three  of  which  are  collinear  and  therefore  constitute  In  six  different  ways  a 
point  conic  and  its  outside  point.  Also  that  any  one  of  these  triangles  and  two  ot 
the  points  A,B.C  form  a  point  conic  left  invariant  by  T^  and  hence  that  T3  leaves 
Invariant  9  different  non-degenerate  point  conies.  If  A,B,C  be  taken  as  the  tri- 
angle of  reference  the  two  transformations  of  type  I3  which  leave  it  invariant  are 

'^z  •     (iXn'=vx.,     and   T3- :     pxJ=ixo 
px./=ix^  iu\\'=rx.^ 

Since  any  triangle  can  be  transformed  into  any  other  triangle  by  a  coUineation 
within  the  PG(2,2-)  it  follows  that  every  coUineation  of  type  I3  is  conjugate  to 
T3  or  T3-.  Since  21 -20 -Ki/^  !..-1120  different  triangles  can  be  chosen  in  PG 
(2,2-)  there  are  1120  conjugate  cyclic  groups' G3  (eye,  L,). 

Type  II.  A  colllneation  To  of  type  II  leaves  invariant  two  real  points  A,B, 
and  a  real  line  /  (distinct  from  AB)  through  one  of  the  points,  say  A.  Two 
lines  fixed  through  A  make  the  transformation  of  lines  through  A  of  period  three. 
One  line  fixed  through  B  makes  the  transformation  of  lines  through  B  of  period 
two. 

T„  is  therefore  of  period  H,  but  only  T,  and  To^=To"'  are  of  type  II.  To" 
and  To*  are  homologies  with  B  for  center  ?nd  /  for  axis  and  To^  Is  an  elation 
with  A  for  center  and  AB  for  axis. 

If  we  select  A  as  the  point  {0,0, r)  B  as  the  point  {1,0,0)  and  the  line  /  as  the 
line  x,  =  r7  we  find  that  any  point  P  not  on  AB  or  /  can  be  transformed  into  any 
other  point  P'  not  on  AB,  /,  PA  or  PB  Taking  P  and  P'  as  {1,1,1)  and  {i,i,o) 
To  is  determined  as 

px/=ixi 
T, :      nx/^=x., 

px/=x.-.-\-x^ 

On  the  line  x,=o  To  interchanges  the  points  {0,1,0)  and  {o,t,i).  It  is  easily 
seen  that  To  or  T.^'  is  conjugate  to  anv  other  transformation  To'  of  type  II 
inPG(2,2-): 

Since  the  invariant  figure  can  be  chosen  in  21  rlO  ■8==1G80  different  ways 
and  for  a  given  point  Pj  on  /  there  are  three  choices  for  P/  it  follows  that 
Goo48o  contains  5040  cyclic  groups  G^  (cyc.II)  each  containing  a  self-conjugate 
cyclic  subgroup  of  order  two  consisting  of  'IV^  (^n  elation)  and  the  Identity,  and 
a  self-conjugate  cyclic  subgroup  of  order  three  consisting  of  To"  and  T./  (hom- 
ologies) and  the  identity.  It  Is  to  be  noted,  however,  that  each  subgroup  of 
order  two  Is  common  to  16   (since  there  are  4  choices  for  B  on  AB  and  4  choices 


14  U.  G.  Mitchell:  Geometry  and 

for  /  through  A)  different  groups  G^  (eye.  II)  and  hence  that  there  are  but  315 
such  subgroups.  Also  that  each  subgroup  of  order  three  is  common  to  lo  different 
groups  Gjj  (eye.  II)  (since  there  are  5  choices  for  A  on  /  and  3  choices  for  the  pairing 
of  h'nes  through  B  in  each  ca;^c)  and  hence  there  are  but  3.3G  different  such  sub- 
groups of  order  three. 

Type  III.  A  transformation  T  of  type  III  leaves  invariant  a  line  /  and  a 
point  A  on  /.  Since  one  line  through  A  is  fixed  the  transformation  of  lines  through 
A  is  parabolic.  T"  is  therefore  an  elation  and  T"*  must  be  the  identical  trans- 
formation. T  is  consequently  of  period  4  and  permutes  four  points,  no  one  on  / 
and  no  three  of  which  are  coUinear,  in  cyclic  order.  Since  the  transformation 
of  four  points  no  three  of  which  are  collinear  into  four  such  points  fully  de- 
termines a  projective  transformation  it  follows  that  a  transformation  T  of  type 
III  is  fully  determined  by  any  four  such  points  which  T  permutes  in  cyclic  order. 

The  collmeation  of  type  III  determined  by  permuting  the  four  points  {i,o,o), 
( i,i,o),{i,0,i),  (/,/,/),  no  three  of  which  are  collinear,  in  cyclic  order  as  named  is 

1":     (jx./=^x^-\-x., 

T  leaves  invariant  the  point  (o,o,i)  and  the  line  x.^=o.  It  is  readily  seen  that  in 
PG(2,2-)  every  collineation  of  type  III  is  conjugate  to  either  T  or  T^. 

Four  noncoUinear  points  A.B.CD  can  be  chosen  in  21 -20 -IG  rO  4  !=2.520  dif- 
ferent ways.  Each  cyclic  order  determines  a  transformation  of  type  III  not  a. 
power  of  any  determined  by  any  other  cyclic  order  and  each  transformation  of 
type  III  permutes  in  cyclic  order  the  points  of  four  different  quadrangles.  It 
follows  therefore  that  there  are  2.')2li -3  4=1800  cyclic  groups  G^  (eye.  Ill)  in 
Geoiso  ^sch  containing  a  self-conjugate  cyclic  subgroup  of  order  two.  It  is  to  be 
noted  that  a  subgroup  of  order  two  is  common  to  G  different  groups  G4(cye.  Ill) 
and  hence  that  there  are  but  315  such  groups. 

Type  If.  Homologies.  The  homologies  in  PG(2.2-)  hx^c  appeared  as  the 
33G  cyclic  subgroups  of  the  201G  G,-  (eye.  I.)  and  the  5040  G,,  (eye.  II).  It 
was  shown  that  the  33G  cyclic  G3  (eye.  IV)  were  conjugate  under  the  group 
GgoiSA-  A  homology,  as  has  been  seen,  is  of  period  three  and  its  path-curves  are  the 
straight  lines  through  the  center.     The  homology 

'J  :     px/=x2 
o.v_.'=a:3 
may  be  taken  as  a  canonical  form. 

Type  I',  hlations.  The  elations  have  appeared  as  315  conjugate  cyclic  sub- 
groups of  order  two  in  both  the  5040  groups  G,.,  (eye.  II)  and  the  1890  groups 
G4  (eye.  III).  Each  elation  is  of  period  2  and  its  path-curves  are  the  straight 
lines  through  the  center.     The  elation 

T:       ;,X/^X^ 


COLLINEATION    GROUPS   OF    PG(2,2-).  15 

which  has  for  center  the  point  (0,0, i)   and  for  axis  the  line  x^^^o  may  be  taken 
as  a  canonical  form. 

§5.      The  Group   of  Determinant   Unity-  — Goqigo- 

Theorem  5,  In  PG  (2,2-)  every  group  G  of  order  N  u'hich  contains  col- 
imeations  of  determinant  not  unity  contains  exactly  N/3  colUne'ations  of  determi- 
nant unity. 

Proof.  It  is  obvious  that  the  coUineations  of  determinant  unity  in  G  form  a 
self-conjugate  subgroup  Gn.  Suppose  n  greater  than  N/3  .  If  T  be  any  collinea- 
tion  in  G  but  not  in  G^  the  products  of  T  and  T"  by  the  n  coUineations  in 
Gn  are  2n  distinct  coUineations  and  G  would  contain  3n>N  distinct  coUinea- 
tions which  is  contrary  to  hypothesis.  Suppose  n  to  be  less  than  N/3  .  G  must 
then  contain  m>  N/3  coUineations  of  determinant  d  where  d  is  either  /  or  r'. 
If  T  be  any  coUineation  in  G  of  determinant  d"  the  products  of  the  m 
coUineations  of  determinant  d  by  T  are  m>N/3  distinct  coUineations  of  determi 
nant  unity  in  G,  contrary  to  supposition.  Since  n  is  neither  less  nor  greater  than 
N/3  it  follows  that  n=  N/3 

The  Group  of  Determinant  Unity.  By  Theorem  5  the  group  Geo48o  has  a  self-con- 
jugate subgroup  of  determinant  unity  of  order  60480/3=20160.  In  §4  it  was  shown 
that  all  coUineations  of  types  I,,, III,  V  and  those  of  type  Iq  of  period  7  and  type 
I  of  period  5  were  of  determinant  unity.  Hence  the  Groupoomo  of  determinant 
unity  contains  the  following  coUineations: 

The    identical    coUineation 1 

All   coUineations   of   type    I, 2240 

Those  coUineations  of  type  Ii  which  are  of  period  5, 

(1-3  of  the  total  number) 8064 

Those  coUineations  of  type  Iq  which  are  of  period  7, 

(3-10  of  the  total  number) 5760 

All   coUineations   of   type   III 3780 

All    coUineations   of    type   V 315 


20160 

The  group  will  be  designated  as  above  by  Gomoo-  It  has  been  proved  that  in  any 
PG(k,p")  the  group  of  all  coUineations  of  determinant  unity  is  the  maximal 
simple  subgroup  of  all  coUineations  in  the  PG(k,p").* 

Theorem  6.  Ei'ery  coUineation  in  Gomoo  c^"  be  obtained  as  a  product' 
of  elations. 

Proof.     Let  T    be    any    coUineation    of    type   I,    determined    by  the    equation 


Cf.  Veblen  and  Bussey,  1.  c,  p.  253  and  Dickson,  I.e.,  p.  87. 


16  U.  G.  Mitchell:  Geometry  and 

TCA.AoAaAJ^AiAoAaAj  where  no  three  of     the     points     A,,A.,A.,,A^,A.     are 
collinear.     Two  elations  Ej  and  E^,  are  ileterniined  by  the  following  equations: 

E,(A,A.A3A,)  =AiA3A,A, 

E,  ( AA.A4A, )  =  A,  A.A,  A, 
such  that  their  product  E2E,=T.  That  E,  and  E^  are  elations  follows  from  the 
facts  that  elations  are  the  only  coUineations  in  PG(2,2-)  of  period  two  and  that 
the  points  Ai,A^,,A.5,A4,A-  are  no  three  collinear.  Since  the  five  points  are  no 
three  collinear  they  form  a  conic  and  since  Eo  interchanges  four  of  the  points  by 
pairs  it  leaves  invariant  point  by  point  the  diagonal  line  of  the  complete  quadran- 
gle of  the  four  points.  By  Corollary  5  of  Theorem  4  this  line  contains  the  fifth 
point  of  the  conic.  Eo,  therefore,  leaves  Ai  invariant  and  it  is  clear  that  EjEj, 
-=  T.  Hence  every  collineation  of  type  L  can  be  obtained  as  the  product  of  two 
elations. 

Let  T,  be  any  collineation  of  type  Ij  of  period  5  determined  by  the  equation 

T,  (A,A,A3AJ  ==A,A3AA- 
where   Ai,A2,A3,A4,Aj    are    five    points   no    three   of    which   are    collinear.      Two 
elations  E/  and  Eo'  are  determined  by  the  equations 

E/(A,AAA,)— A.AAA. 

E/  ( A3A3AA. )  =A,AAA2, 
such  that  Eo'E/^T,.     That  E/  is  an  elation  leaving  A.,  invariant  and  that  "E^ 
is  an  elation  leaving  A,   invariant  follows  by  the  reasoning  given  above  to  show 
that  E,  was  an  elation  leaving     A,   invariant.        Hence  every  collineation  T^  of 
type  I,  of  period  five  can  be  obtained  as  the  product  of  two  elations. 
The  transformation 

T^:     px./=x^-\-x^ 

px/=x,-\-x.,^x.^ 
is  of  type  I^,  of  period  7.     It  is  found  that  To=  T^E  where  E  is  an  elation, 


E: 
md  Ti  is  of  type  I^  of  period  five 

T, 


DX./=Xn 

pX./=X^-\-X3 


rx./=x^ 

pX./=X.,-\-X., 

But  since  every  collineation  of  type  Ij  of  period  five  can  be  obtained  as  a  product 
of  elations  and  T,j  or  one  of  its  powers  is  conjugate  to  every  collineation  of  type 
I(,  of  period  seven  within  the  PG(2,2-)  it  follows  that  every  collineation  of  type 
lo  of  period  seven  can  be  obtained  as  a  product  of  elations. 

Let  S  be  any  collineation  of  type  HI  determined  by  the  equation 

S(A,A,A3AJ=AAA4A„ 

where  no  three  of  the  points  A,,A2,A3,A4  are  collinear.  Any  transformation  so 
determined  must  be  of  type  HI  because  m  PG(2,2-)   transformations  of  type  HI 


COLLINEATION    GrOUPS   OF    PG('2,2-).  17 

and  no  others  are  of  period  four.     Then  S=E2Ei  where  E^  and  Eo  are  elations 
determined  by  the  equations 

Ei(AAAA)  =  AAA3A,, 
E.CA.A.AgAJ  =  AoAAa's- 

El  and  E2  are  again  necessarily  elations  because  elations  are  of  period  two  and 
the  points  are  no  three  coUinear. 

Since  every  collineation  not  an  elation  in  Gooieo  must  be  of  type  I3,  I^  (of  pe- 
riod 5),  Ip  (of  period  7),  or  III  the  theorem  is  established. 

Theorem  7.     In  PG(2,2-)   if  a    group    Ga    of    determinant    unity    be  trans^ 
itive  on  all  points  and  lines  of  the  plane  and  contain  a  single  elation  it  contains 
all  elations.  • 

Proof.  In  PG(2,2-)  three  and  but  three  elations  have  the  same  center  and 
axis  since  there  are  but  three  ways  in  which  the  four  points  other  than  the  center 
on  an  invariant  line  can  be  paired.  The  theorem  will  follow,  therefore,  if  it  can 
be  shown  that  if  Ga  contain  a  single  elation  it  must  contain  elations  such  that  for 
any  given  line  /  and  point  P  on  /  there  are  three  elations  in  Ga  having  P  for 
center  and  /  for  axis. 

From  the  transitivity  of  Ga  it  follows  that  Ga  must  contain  transforms  of  the 
given  elation  such  that  every  point  in  the  plane  is  the  center  and  every  line  the 
axis  of  at  least  one  elation.  Also  the  order  of  Ga  must  be  a  multiple  of  21  and 
therefore  Ga  must  contain  a  collineation  of  period  3.  Since  the  only  collineations 
in  PG(2,2")  of  determinant  unity  and  period  three  are  of  type  I3  it  follows  that 
Ga  must  contain  collineations  of  type  I3  such  that  every  point  in  the  plane  is  a 
vertex  and  every  line  of  the  plane  is  a  side  of  the  invariant  triangle  of  at  least 
one  collineation  of  type  I3. 

For  the  given  line  /,  then,  there  is  in  Ga  an  elation  E  having  /  for  axis  and  a 
collineation  T  of  type  I3  having  /  for  an  invariant  line.     Four  cases  may  arise, 
(a).     P  may  be  the  center  of  F  and  an  invariant  point  of  T. 

Since  T  leaves  invariant  a  point  which  E  transforms  they  cannot  be  commuta- 
tive  and   hence  TET"'    and  T-ET"-  are  the  other  two  elations  having  P  for  cen- 
ter and  /  for  axis, 
(b).     P  may  be  the  center  of  E  and  not  an  invariant  point  of  T. 

Let  A  and  B  be  the  invariant  points  on  /  of  T.  A  must  be  the  center  of  some 
elation  E^.  If  /  be  not  the  axis  of  E^  we  have  E^TEi'^^Ti  a  collineation  of 
tjpe  I3  having  A  and  some  point  B'  different  from  B  on  /  for  invariant  points.  By 
transforming  Ti  through  the  power  of  T  which  transforms  B'to  P  the  case  is  reduced 
to  case  (a).  A  similar  argument  applies  to  the  point  B.  If  neither  A  nor  B  be 
the  center  of  an  elation  whose  axis  is  not  I,  by  case  (a)  Ga  must  contain  all 
elations  having  A  or  B  for  center  and  /  for  axis.  The  three  elations  Ei,  E,,  E3 
having  A  for  center  and  /  for  axis  form  with  the  identity  a  group  since  the 
product  of  any  two  of  them  is  an  elation  having  /  for  axis  and  A  for  center. 
Similarly  the  three  elations  ^^'X/^^i  having  B   for  center  and   /  for  axis  form 


18  U.  G.  Mitchell:  Geometry  and 

a  group.  The  nine  products  EjEj'  are  all  distinct  since  if  EiEj'=E,;Ei'  it  fol- 
lows that  Ej'E/=E,Ek  which  cannot  be  true.  Moreover,  every  EiEj'  is  an 
elation  having  /  foi  axis  since  it  is  of  determinant  unity,  leaves  fixed  every  point 
of  /  and  can  not  be  the  identity.  Since  the  nine  elations  Ei  Ej'  are  all  distinct  and 
iiave  /  for  axis  they  include  the  three  elations  having  P  for  center  and  /  for  axis, 
(c).  P  may  not  be  the  center  of  E  and  may  be  an  invariant  point  of  T. 
P  must  then  be  the  center  of  some  elation  E'  having  some  other  line  than  /  for 
axis.  The  transforms  of  E  through  E',  T  and  T-  give  elations  such  that  every 
point  of  /  other  than  P  is  the  center  of  an  elation  having  /  for  axis.  Since  the  lines 
through  P  can  be  interchanged  by  pairs  in  three  ways  only,  the  product  of  some 
two  of  these  four  elations  is  an  elation  with  P  for  center  and  /  for  axis.  This  case 
is  thereby  reduced  to  case   (a). 

(d).  P  may  be  neither  the  center  of  E  nor  an  invariant  point  of  T.  If  C,  the 
center  of  E,  be  not  an  invariant  point  of  T  by  transforming  E  through  T  or  T^ 
.'whichever  transforms  C  to  P)  the  case  is  reduced  to  case  (b).  If  C,  the  center  of 
E,  be  one  of  the  invariants  points  of  T  we  may  transform  E  through  E^,  the  elation 
having  P  for  center  and  some  other  line  than  /  for  axis,  and  obtain  an  elation  Eg 
having  some  other  point  on  /  for  center.  If  E,  have  for  center  the  other  invariant 
point  of  T  the  product  EE,  is  an  elation  "Eg  whose  center  is  not  one  of  the  invariant 
points  of  T.  The  transforming  of  E,  or  E3  through  T  or  T-  then  reduces  this 
case  as  above  to  case  (b),  and  completes  the  proof  of  the  theorem. 

Definition  of  Figure.  In  PG(2,'2")  a  point  figure  is  defined  as  any  set  of  m 
points  where  in  is  any  positive  integer  less  than  2-"-|-2"-[-l.  Similarly  a  line  figure 
consists  of  any  m  lines.  The  term  figure  is  used  to  refer  to  either  a  point  figure  or 
a  line  figure.  A  real  figure  in  PG(2,2")  is  a  figure  all  of  whose  points  and  lines 
belong  to  the  PG(2,2"). 

It  is  obvious  that  any  subgroup  of  Ggo^s,  which  leaves  invariant  no  real  figure  is 
transitive  on  all  points  and  lines  of  the  plane. 

Theorem  8.  There  is  no  subgroup  of  Goqico  which  does  not  leave  invariant  a 
real  figure  on  an  imaginary  triangle. 

Proof.  Any  such  subgroup  Gk  can  contain  no  elation,  for  by  Theorem  7  if  Gk 
contained  a  single  elation  it  would  contain  all  elations  and  hence,  by  Theorem  6,  all 
collineations  in  Gjoieo-  Also  Gk  can  contain  no  coUineation  of  type  III  since  the 
square  of  a  type  III  is  an  elation. 

Suppose  Gk  to  contain  a  coUineation  T,  of  type  I,  and  let  its 
center  be  designated  P,.  As  was  seen  in  the  proof  of  theorem  7,  since 
Gk  is  transitive  and  of  determinant  unity  it  contains  some  coUineation 
T3  of  type  I3  which  leaves  Pj  invariant.  Let  /j  and  L  be  the  two  lines  through 
Pi  left  invariant  by  T3.  Since  T,  is  of  period  5  on  the  lines  through  Pj  some  power 
of  Tj,  say   T^""    transforms  l^    to  L.  Let  1^,^,1^    be  the  lines  into  which  T^"" 

transforms  L^z^h  respectively.  Some  power  of  To,  say  T3",  produces  among  the 
lines  through  P,  the  transformation  (/,)  (/.)  {I  J  J:,)-  Hence  the  product 
"1",-™T3"  produces  among  the  lines  through  P^  the  transformation   (/1/3)  (4^4  )  (^5) 


COLLINEATION    GrOUPS   OF    PG(2,2-),  19 

The  collineation  T^-^T,"  leaves  invariant  the  point  P,  and  a  single  line  /.  through 
P^.  Such  a  collineation  must  be  of  type  III  or  an  elation.  Hence  Gk  can  contain 
no  collineation  of  type  Ij. 

Since  the  onl\'  other  collineations  of  determinant  unity  are  of  type  I3  (of  period  3) 
and  type  I,,  (of  period  7)  Gk  can  contain  only  collineations  of  these  two  types. 
Since  20160^2" '3- -5 -T  and  the  order  of  Gk  must  be  divisible  by  21  the  only  possible 
orders  for  Gk  are  21  and  63.  But  as  a  consequence  of  Sylow's  Theorem*  any  group 
of  order  21  or  03  must  contain  a  self-conjugate  cyclic  subgroup  of  order  7  since  the 
order  of  the  group  can  be  written  in  the  form  7m(l-)-7k)  where  7m  is  the  order 
of  the  largest  group  within  which  the  cyclic  subgroup  of  order  7  is  self- 
conjugate  and  l-|-7k;  is  the  number  of  cyclic  subgroups  of  order  7.  For  order  21 
the  only  possibility  is  k;==0  and  m==3  and  for  order  63  k  =0  and  m^9.  In 
PG(2,2-)  the  only  possible  cyclic  group  of  order  7  is  a  G-  which  leaves  invariant 
an  imaginary  triangle  Fi.  But  if  the  G-  be  self-conjugate  within  the  Gk  every 
collineation  in  Gk  must  leave  invariant  the  imaginary  triangle  F;.  Hence  there 
is  no  subgroup  Gu  of  Goqioo  which  docs  not  leave  invariant  either  a  real  figure  or 
an  imaginary  triangle. 

Theorem  P.  There  is  no  subgroup  of  Geo^so  except  G20160  '^i-'hich  docs  not 
leave  invariant  a  real  figure  or  an  imaginary  triangle. 

Proof.  If  any  subgroup,  say  Gn,  exist  it  must  contain  a  self-conjugate  sub- 
group Hn  of  determinant  unity  which  leaves  invariant  no  real  figure  or  imaginary 
criangle  contrary  to  theorem  8. 

§6.      The   Group  Leaving  Invariant  an   Imaginary    Triangle — Gq^. 

Theorem  10.  The  group  of  all  collineations  in  PG(2,2-)  ivhich  leave  invct- 
riant  a  given  imaginary  triangle  Fj  is  of  order  63, 

Proof.  Let  the  group  be  designated  Ga-  In  §4  it  was  shown  that  if  a  collin- 
eation leave  fixed  one  vertex  of  Fj  it  leaves  fixed  every  vertex  of  ¥ ,.  Hence  every 
collineation  leaving  Fi  invariant  must  either  permute  the  vertices  of  Fi  in  cyclic 
order  or  leave  each  vertex  fixed.  It  was  also  shown  in  §  4  that  there  are  exactly 
21  collineations — the  21  powers  of  a  type  I,,  of  period  21— which  leave  each  vortex 
of  an  imaginary  triangle  Fi  fixed.  That  there  can  not  be  more  than  21  such  col- 
lineations follows  from  the  fact  that  ?  collineation  is  fully  determined  by  the 
leaving  fixed  of  each  vertex  of  an  imaginary  triangle  and  the  transformation  ot 
one  real  point  into  another  real  point. 

There  can  not  be  more  than  21  collineations  permuting  the  vertices  AiBiCi  of  Fj  in 

a  given  cyclic  order  (AjBiCi).     For  suppose  S,,  S.;,,  S^, Sn  to  be  n  such 

collineations  where  n>21.     Then  if  T  be  a  collineation  permuting  Ai.Bi.Ci   in 

the  order  (AjCiBi)   there  are  within  G,  n>21  collineations  TS^.TS.,  TS.^ 

,TSn,   distinct   from  each  other  and  each  leaving  every  vertex  Ai.Bi.Q 


*  See  BuRNSiDE,  Theory  of  Groups,  p.  94. 


20  U.  G.  Mitchell:  Geometry  axd 

fixed,  contrary  to  the  hypothesis  that  G^  contains  but  21  collineations  leaving  each 
vertex  of  Fi  fixed. 

That  there  exists  a  group  of  order  C3  leaving  Fi  invariant  is  shown  by  consid- 
eration of  the  transformations 

px/=ix^-\-x^  px,'=x^-\-x^ 

[  (,:      px./=x^-\-ixn  and  T3 :       px./^x^ 

px/=^x._-j-rx^  px/=Xo-\-x^ 

To  is  of  type  I^,  of  period  21  and  T3  is  of  type  I,  of  period  3.  T^  leaves  fixed 
each  vertex  Ai^{i,v-'jv"''),  Bi^{i,v*^,v  ^'^)'Ci^  {i,v''*.v'')  [where  v  is  a  prim- 
itive root  of  the  GF(2^)]  of  Fj,  and  I3  permutes  these  vertices  in  the  order 
(AiBiCi).  Ty  and  T3,  therefore,  generate  u  group  of  order  63  leaving  invariant 
the  imaginary  triangle  Fj.  Since  it  was  shown  in  §3  that  Fj  can  be  transformed  into 
iiny  other  imaginary  triangle  Fj  by  a  coUineation  within  the  PG(2,2-),  there  is  a 
group  of  order  63  leaving  invariant  an}   such  triangle. 

Theorem     11.     The  only  groups  in  PG{2,2-)    nhich   leave  invariant  an   im- 
aginary triangle  Fj  are  the  follouing: 

A.  Groups  leaving  each  vertex  of  F,  fixed. 

Z.J  cyclic  group  G^icycl^)  of  collineations  of  type  /^  of  period  J. 

b.  J  cyclic  group  G-{cyc.  I^,)   of  collineations  of  type  If,  of  period  7. 

c.  J  cyclic  group  GoiCo'^-  -^o)   0/  coVineations  of  type  /^  of  period  21. 

B.  Groups  permuting  the  vertices  of  F,. 

a.  A  cyclic  group  G^{cyc.  /„)   of  collineations  of  type  /^  of  period  3. 

b.  A  cyclic  group  Gz{cyc.  I^)  of  collineations  of  type  /^,  of  period  3. 

c.  An  Abelian  group  Gg  leaving  invariant  also  a  real  triangle,  and  contain- 
ing besides  the  identity  6  collineations  of  type  I^  of  period  3  and  2  col- 
lineations of  type  /g. 

d.  A  self-conjugate  group  Gn^  of  determinant  unity  containing  besides  the 
identity  6  collineations  of  type  7,^  of  period  7  and  14  collineations  of  type  I^. 

t.  A   group   Gc3   of  all  collineations   in   PG{2.2-)    ichich    leave   Fj    invariant 
containing  besides  the  identity  6  collineations  of  type  1 ,,  of  period  7,  30  of 
type  /„  of  period  3,  12  of  type  /„  oi  peiiod  21,  and,  14  of  type  I-,. 
Proof.     The  existence  of  the  group  G,,,  of  all  collineations  in  PG(2,2-')  leaving 
1'  1  invariant  was  shown  in  the  proof  of  the  preceding  Theorem.     The  existence  of 
the  cyclic  subgroups  is  obvious  and  the  existence  of  the  Go,  of  determinant  unity 
follows  from  Theorem  5.     The  group  G..  is  a  Sylow  subgroup  and  that  it  is  Abe- 
lian follows  from  the  fact  that  its  order  is  the  square  of  a  prime.*     To  establish 
the  Theorem  it  is  only  necessary  to  show  further  that  every  subgroup  of  the  G^s  of 
all  collineations  leaving  Fi  invariant  is  one  of  the  kinds  enumerated  above.     The 
only  possible  orders  for  such  subgroups  ar^  3,  7,  9,  and  21.     All  subgroups  of  order 
3  or  7  must  be  among  the  cyclic  groups  enumerated  above  since  3  and  7  are  primes. 
A  group  of  order  9  must  be  Abelian  and  by  Theorem  5  must  contain  a  GgCcyc.  I3) 


*  Cf.   BURNSIDE,  1.   c,  p.   63. 


COLLINEATION    GrOUPS   OF    PG(2,2-).  21 

of  determinant  unity.  But  no  such  Gg  can  contain  more  than  one  G3(cyc.  I3)  ; 
for  if  T,  and  To  be  two  collineations  of  type  I3  which  do  not  belong  to  the  same 
^^3 (eye.  I3)  the  product  of  T^  by  the  power  of  T,  which  permutes  the  vertices 
of  Fi  in  inverse  order  is  a  coUineation  of  determinant  unity  leaving  each  vertex  of 
Fj  fixed  and  therefore  of  type  I,,  of  period  7.  Hence  every  subgroup  of  G^^  of 
order  0  contains  a  self-conjugate  G3(cyc.  I3)  and  leaves  invariant  a  real  triangle. 
A  subgroup  of  G63  of  order  21  must  be  the  direct  product  of  a  G3  and  a  G^.  Since 
the  G7  must  be  a  G^  (eye.  I,,)  and  the  G.,  must  be  either  a  G3(cyc.  I,,)  or  a  G3 
(eye.  I3)  every  such  subgroup  must  be  either  a  G2i(cyc.  I^)  or  a  Go^  of  determinant 
unity  and  therefore  one  of  the  kinds  enumerated  in  the  Theorem. 

§  7-  Invariant  Real  Figures  and  Their  Groups. 

It  has  now  been  shown  that  every  sub;2:roup  of  the  G,5048o  except  the  self-conju- 
gate Go„ioo  of  determinant  unity  leaves  invariant  a  real  figure  or  an  imaginary 
triangle,  and  every  group  which  leaves  invariant  an  imaginary  triangle  has  been 
determined.  Accordingly  we  next  take  up  the  question  of  determining  what  real 
figures  can  be  the  invariant  figures  of  groups  in  PG(2,2-)  and  what  group  or 
groups  leave  each  invariant.  In  determining  these  groups  it  is  sufficient  to  con- 
sider point  figures;  for,  since  a  coUineation  in  the  plane  is  self-dual,  corresponding 
to  every  group  which  leaves  invariant  an  n-line  figure  there  is  a  group  of  the  same 
order  which  leaves  invariant  the  dual  //-point  figure.  Abstractly  considered  the 
two  groups  are  identical.  Furthermore,  in  PG(2,2'-)  it  is  sufficient  to  consider 
point  figures  in  which  the  number  of  points  n  is  less  than  11,  for  if  ;/  i^  11  the 
point  figure  consisting  of  21-n  (or  some  lesser  number)  can  be  taken  as  the  inva- 
riant figure  of  the  group. 

In  this  section  will  be  determined  all  groups  which  leave  invariant  real  point- 
figures  whose  points  are  not  all  collinear  and  which  leave  no  point  fixed  under 
all  transformations  of  the  group.  To  obtain  all  such  groups  it  is  only  necessary  to 
determine  for  each  value  of  n  from  «  =  10  to  n  =  3  all  groups  which  are  trans- 
itive on  all  points  of  the  «-point  figure ;  for,  if  such  a  group  be  not  transitive  on 
all  points  of  an  w-point  invariant  figure  it  must  appear  as  a  group  which  is  trans- 
itive on  an  ///-point  figure  where  3g///</;. 

A  group  which  leaves  invariant  an  //-point  figure  also  leaves  invariant  an  as- 
sociated line  figure  each  line  of  which  contains  the  same  number  of  points;  for, 
every  line  containing  k  of  the  n  points  can  be  transformed  by  a  coUineation  within 
the  group  into  some  line  through  each  of  the  other  Ti-k  points  and  hence  each 
of  the  n  points  lies  on  at  least  one  line  containing  k  of  the  n  points.  It  is,  of 
course,  obvious  that  every  transform  of  a  line  which  contains  k  of  the  n  points 
must  also  contain  k  of  the  n  points  if  the  //-point  figure  is  invariant  under  the  group. 
It  follows  by  the  same  reasoning  that  through  each  of  the  //  points  there  must  be 
the  same  number  of  lines.     Hence  the  figure  made  up  of  an  //-point  and  its  asso 


22 


1j.  G.  Mitchell:  Geometry  axd 


dated  m    line    may    be    called    a    configuration'^    and    represented    by  the    symbol 


i  "   ^ 

k   in 


where  n  is  to  indicate  the  total  number  of  points,  ///  the  total  number 
of  lines,  k  the  number  of  paints  on  each  line  and  /  the  number  of  lines  through 
each  point  of  the  configuration.  In  sucli  a  configuration  the  points  and  lines  are 
so  related  that  nl  =  km,  and  1  -\-  l{k-l)'^n.  Also,  since  in  PG(2,2-)  not  more 
than  G  points  can  be  chosen  such  that  no  three  are  coUinear  (Cor.  4,  Theorem  4) 
if  n  >  6,  ;t  <{;  3.  Since  not  more  than  6  lines  can  be  chosen  such  that  no  three  arc 
concurrent  it  follows  that  if  m  >6  either  /<};3  or  n<i^'ik. 

By  making  use  of  the  above  relations  and  the  fact  that  whenever  m<in  or 
'21 — 771  <n  it  follows  by  duality  that  the  same  group  must  appear  as  a  group  leaving 
Invariant  a  lesser  number  of  points,  we  find  that  the  possible  configurations  in 
JPG (2,2-)   reduce  to  the  following: 


(a) 


(d) 


(g) 


(J) 


|10 

3   1 

1   3 

10  1 

p8" 

3    1 

1   3 

s  1 

6 

4   1 

2 

12  1 

1   5 

4   1 

1   - 

10  1 

Fu 


^  sv-i 


(b) 


(e) 


(h) 


(k) 


1   9 

-t   1 

1   3 

12  1 

1   7 

3   1 

1  3 

'    1 

I   6 

3 

1   2 

9 

1   5 

^   1 

1     0 

1    - 

5   1 

=F     ' 


=F. 


(c) 


(f) 


vi) 


(1) 


1   9 
1   3 

3 
9 

1   ^ 
2 

5  ! 

15  1 

1   ^> 

1   2 

2   1. 
6   1 

\ 

'     4 
2 

3 
G 

=F    ' 


-Fo„ 


=F,. 


(m) 


=F    ' 

A  4'2 


(n) 


-  I  =F 
3  I       ^=*'=- 


It  is  observed  that  a  group  which  leaver,  invariant  an  Fi,j  also  leaves  invariant 
the  figure  made  up  of  the  remaining  points  and  lines  of  the  plane.  This  figuv 
will  be  called  the  residual  figure  and  referred  to  as  Ri,j. 

We  will  consider  these  configurations  in  order. 


U) 


3 
10 


10 


F 

■•■  io>a 


Consider 


pomt 


of 


the       three     lines     l^.l-.J^     of 


*  Cf.  Veblen  and  Young,  Projective  Geometry,  Vol.  I,  pp.  38-39. 


COLLINEAIION    GrOUPS   OF    PG(2,2-), 


23 


Fj„,3  which  pass  through  P.  On  l^,i,,l.  are  7  points  of  F,,,,^  and  hence  there  are 
three  points  Pi,?.,?.,  of  Fj,,,,  not  on  any  of  the  lines  liJ.jJ^-  This  necessitates 
either  that  6  lines  I^J-.'^a'  PPiiPPo.PPs  pass  through  P  or  that  two  or  more  of  the 
points  P^jPo.Pg  are  coUinear  with  P.  Since  neither  of  these  conclusions  is  allow- 
able under  our  hypotheses,  Fio,3  is  not  a  possible  configuration  in  PG(2,2'-). 

(b) 


4 
12 


On  each  line  of  F<,,3  must  be  two  points  which  do  not  belong  to  Fo,3,  Let  any 
line  of  Fj,,3  be  chosen  as  the  line  x^^o  and  the  two  points  on  it  which  do  not 
belong  to  F,,,,  as  the  points  12  {o,o,i)*  and  17  {o,i,o).  Then  the  points  0 
{o,i,i),  10  {o,i,i),  and  18  (o.i.i)  on  x,^=o  must  be  points  of  F9,3.  Through  0 
passes  one  and  but  one  line  which  does  not  belong  to  F,j,3.  Let  the  point  of  inter- 
section of  this  line  with  the  similar  line  through  10  be  chosen  as  the  point  4 
{i,o,o).  Neither  of  these  lines  can  contain  any  other  points  of  F.j,3  than  0  and 
-0,  respectively,  because  the  other  three  lines  through  either  point  contain  six  other 
points  of  Fg,3  which  added  to  the  three  points  on  x^=o  gives  the  total  nine  points 
of  F,,,3.  The  point  4  is  therefore  not  a  point  in  F,,,,.  Now  no  line  through  12 
which  does  not  pass  through  i  can  be  a  line  of  F<„3  since  such  a  line  contains  at 
least  three  points  (12  and  its  two  intersections  with  the  lines  from  4  to  0  and  10, 
respectively,)  not  in  Fr,,3.  A  similar  argument  applies  to  the  point  17.  But  every 
line  of  Ff,,3  passes  through  scime  point  of  Xj^=o  and  but  nine  besides  x^=o  pass 
through  the  points  0,  10,  18.  Hence  the  lines  a-2=o  and 
x.^=o  are  lines  of  F,,,,  and  the  nine  points  of  Fo,3  lie 
three  by  three  on  the  sides  of  the  triangle  of  refer- 
ence On  the  side  x.j=o  are  the  points  6  (i.o.i), 
11  {i,o,i),  15  {i,o,i),  and  on  Ar3=o  are  3 
{i,i,o),  7  {i,i,o),  19  (/,  i,o).  A  reference 
to  the  table  of  alignment  (p.  3)  shows  that 
these  nine  points  are  collinear  by  threes 
on  nine  other  lines  as  shown  in  the 
accompanying  figure  (Fig.  3). 
Since  x^=o  was  chosen 
<is  atiy  line 


Fii^ure  3 


*  Numbers  printed  thus,  12,  17,  etc.,    refer    to    the    numbers  assigned  to  points  with  certain 
coordinates,  as  given  in  the  talile  of  alignment  on  p.    3 


24  U.  G.  Mitchell:  Geometry  and 

in  Fj„3  it  follows  that  through  each  point  not  belonging  to  Fg.,  pass  two  lines  of 
F,,.3  and  three  lines  not  belonging  to  F.03,  and  each  une  not  belonging  to  Fg,, 
contains  one  and  but  one  point  of  Fr,.,. 

Having  found  that  Fa,3  is  a  possible  configuration  in  PG(2,2-)  we  next  pro- 
ceed to  determine  what  coUineations  can  leave  it  invariant. 

No  line  of  Fg.3  can  be  the  axis  of  an  elation  leaving  Fj„3  invariant,  for  an 
elation  interchanges  all  lines  not  invariant  by  pairs.  Hence  not  more  than  one 
point  of  F9,3  can  be  invariant  under  an  elation.  But  since  an  elation  interchanges 
all  points  not  invariant  by  pairs  at  least  one  of  the  nine  points  of  F^,^  must  be 
invariant  under  an  elation  which  leaves  Fj„3  invariant.  If  any  point  of  Fg,3  is  to 
be  the  center  of  such  an  elation  the  axis  of  the  elation  must  be  the  one  Ime  through 
the  point  which  contains  no  other  point  of  Fg.3,  that  is,  the  axis  must  be  the  line 
joining  the  point  to  the  opposite  vertex  of  the  triangle  of  reference.  An  elation 
having  such  a  center  and  axis  and  interchanging  the  other  two  vertices  of  the 
triangle  of  reference  must  leave  Frj,3  invariant  since  the  nine  point«-  of  Fj,,3  lie 
three  by  three  on  the  sides  of  the  triangle  of  reference.  It  is  obvious  that  there 
exists  one  and  but  one  such  elation*  for  each  point  in  Fy,3.  Moreover,  no  point 
of  the  residual  figure  R9,3  can  be  the  center  of  an  elation  leaving  F^,^  invariant 
lor  through  such  a  point  pass  two  lines  of  F^.g  upon  each  of  which  are  three  points 
cf  Fg.g  which  could  not  be  interchanged  by  pairs.  There  are.  therefore,  nine  and 
hut  nine  elations  in  PG(2,2-)  which  leave  F,„3  invariant. 

If  T  be  a  transformation  of  type  I,  »vhich  leaves  F,„3  invariant  no  point  P  of 
Fj,,3  can  be  a  vertex  of  its  invariant  triangle ;  for  at  least  one  of  the  invariant  lines 
through  P  would  have  to  be  a  line  of  F,,.^  and  on  that  line  a  point  of  Fj,,3  would 
be  transformed  into  a  point  in  K,,,3.  It  any  point  in  R,j.3  can  be  a  vertex  of  the 
invariant  triangle  of  T  the  two  lines  through  it  belonging  to  Fg.3  must  be  the  two 
invariant  lines  through  the  point,  since  otherwise  at  least  one  of  them  would  be 
transformed  into  a  line  not  in  R,,.;^.  The  other  two  vertices  of  the  triangle  must 
be  the  two  other  points  on  these  lines  which  do  no*"  belong  to  Fy,3.  Since  the  line 
joining  these  two  points  is  a  line  of  F.,.,  the  points  of  Fg,3  are  three  by  three  on 
the  sides  of  the  triangle  and  hence  1'  and  T-  leave  F9.3  invariant.  Since  but  four 
such  invariant  triangles  can  be  selected  irom  the  twelve  points  of  R9,3  there  are 
eight  and  but  eight  coUineations  of  type  F  which  leave  F^.g  invariant. 

No  transformation  of  type  I^  of  period  five  or  fifteen  can  leave  F9,3  invariant 
on  account  of  its  period.  A  collineation  of  type  I3  of  period  three  is  an  homolog}'. 
If  an  homology-  H  leave  Fg.g  invariant  it  can  not  have  a  point  of  F9,3  for  center 
since  on  one  of  the  invariant  lines  a  point  of  F9.3  would  be  transformed  into  a 
point  of  R9,3.  If  any  point  P  of  Ro.y  can  be  the  center  of  H.  through  P  pass 
two  and  but  two  lines  of  F9.3  and  the  axis  of  H  must  be  the  line  /  joining  the  two 
points  of  R9,3  which  lie  on  these  two  lines.     Since  /  contains  the  other  ti^ree  points 


For  example,  if  0   be  chosen  as  the  point,  the  elation  must  be       px/-=l 

px./=i 


COLLINEATION    GROUPS   OF    PG(2,2-).  25 

of  Fgts  a  homology  having  P  for  center  and  /  for  axis  must  leave  Fg,3  invariant. 
For  each  of  the  twelve  points  of  Rg,3  there  are,  then,  two  and  but  two  homologies 
leaving  F,„.^  invariant.  Accordingly,  there  are  twenty-four  homologies  leaving 
Fy,3  invariant. 

If  the  homology  H  having  P  for  center  and  /  for  axis  be  multiplied  by  an 
elation  E  leaving  F,,,.  invariant  and  having  some  point  Q  on  /  for  center  a  col- 
lineation  To  is  obtained  which  transforms  Fg,3  into  itself  and  leaves  invariant  the 
points  P  and  Q  and  the  line  /.  Since  To  is  of  determinant  not  unity  and  is  of 
period  2  on  /  and  period  3  on  the  line  PQ  it  must  be  a  collineation  of  type  11. 
Since  there  are  three  and  but  three  choices  for  the  point  Q  on  /  there  are  six  and 
but  six  collineations  of  type  II  having  P  for  center  which  leave  F9,3  invariant. 
But  every  collineation  of  type  II  is  of  period  six  and  has  for  its  square  a  homology 
and  for  its  cube  an  elation.  Hence  every  collineation  of  type  II  which  leaves  Fg,^ 
invariant  must  be  the  product  of  a  homology  and  an  elation  each  leaving  Fg,3 
invariant  and  therefore  related  as  were  PI  and  E  in  obtaining  To  above.  Since 
the  only  points  which  can  be  the  centers  of  homologies  leaving  Fg,3  invariant  are 
the  twelve  points  of  R,,,,  and  each  homology  and  its  square  can  be  combined  with 
three  different  elations  there  are  seventy-two  and  but  seventy-two  collineations  of 
type  n  leaving  Fg,3  invariant. 

If  Fg,3  can  be  left  invariant  by  a  collineation  Tg  of  type  III,  T3  must  have  the 
same  center  and  axis  as  some  elation  since  T3-  is  an  elation.  Taking  0  for  center 
and  0  4  for  axis  we  find  that  there  are  six  and  but  six  collineations  T3,T3',  T3", 
and  their  cubes,  of  type  III  which  leave  Fg,3  invariant  and  have  this  center  and 
axis.  This  corresponds  to  the  fact  that  there  are  only  three  ways  in  which  the  four 
lines  of  Fg,3  through  0  can  be  interchanged  by  pairs.  The  transformations  T3, 
T./,  and  T3"  are 

px/=Xj^-\-rx.^-{-ix^  px/=x^-\-ix.2-\-x.,  px/==ixi^-\-ix.-,-\-x^ 

T3:    px./^ix.^-\-ix.^^ix^  T/ :px/=i-x-^^-\-ixo-\-ix^    To":  px/=ix^-\-x^-{-ix^ 

px./=rx.^^-\-x.^-{-ix2  px./=x^-\-X2-\-ix^  P-*"3'^^'"'*"i~l~'^2"t"-*"3 

That  these  collineations  leave  Fg,3  invariant  is  more  readily  seen  when  they  are 
written  in  the  form  (points  of  F,,,,  in  italics)  : 

T3=(o)  (  ^  II  15  7)  (^  18' ig  10)  (1  10)  (4  14)  (5  12  9  17)  (2  13  8  20) 
T3'=(o)  (J  19  15  6)  (7  10  II  18)  (1  14)  (4  16)  (5  13  9  20)  (2  17  8  12) 
T3"=(o)    {3  18  15  10)    {6  7   19  11)     (1  4)    (14  lU)    (5  2  9  8)    (12  20  17  13) 

Since  these  collineations  are  not  commutative  with  the  collineations  of  type  I3 
and  the  elations  which  change  the  point  0  into  points  on  the  other  sides  of  the 
triangle  of  reference  it  is  clear  that  the  total  group  of  collineations  leaving  Fg,3 
invariant  must  contain  six  collineations  of  type  I3  for  each  point  of  F9,3  or  alto- 
gether 54  such  collineations.  In  fact,  it  is  obvious  that  if  any  other  center  and  axis 
than  0  and  0  4  respectively  had  been  selected  six  and  but  six  collineations  of  type 
III  leaving  Fg,3  invariant  and  having  that  center  and  axis  could  have  been  deter- 
mined, provided  that  the  center  and  axis  selected  were  the  center  and  axis  of  some 
elation  leaving  Fq,3  invariant. 


26  U.  G.  Mitchell:  Geometry  and 

No  transformation  of  type  I^,  of  period  21  or  7  can  leave  F^.g  invariant  on  ac- 
count of  its  period.  If  a  transformation  T„  be  of  type  I^,  of  period  three  it  per- 
mutes all  points  of  the  plane  by  triangles.  As  we  have  seen,  there  are  four  trian- 
gles each  of  which  has  for  vertices  points  of  Rrj.->  and  all  other  points  on  its  sides 
points  belonging  to  Fj,,,.  If  Tq  leave  Fj,..  invariant  it  must  transform  this  set  of 
triangles  into  themselves,  either  by  permuting  the  vertices  of  one  of  the  triangles 
among  themselves  or  by  transforming  one  triangle  wholly  into  another.  Selecting 
one  of  these  triangles,  say  4  12  17,  we  determine  all  the  transformations  of  type 
1(,  of  period  three  which  permute  its  vertices  in  the  order  (4  12  17)  and  find  that 
there  arc  the  six  following: 

-*■  01       -'-  i'2      -'-  03      ■•■  04        ^  05        ^  06 

px^'=  Xr,  Xn  ix^  Xr,  Xn  i'x^ 
px./==  AT,  \x^  x^  jr,  vx^  x^ 
px/=ix^       x^       A-j      ixj^        Xj^      Xj^ 

Writing  these  in  the  form  (points  of  F^,,  in  italics) 

T„i=(4  12l7)  (oigii)  (3  15  lo)  {6  iS  7)   (12  9)   (5  1.3  8)   (14  20  Ifi) 
T,.=  (4  12  17)  {07  I5){3  II  18)  {6  10  19)    (116  5)  (2  14  13)  (8  9  20) 
To3=(4  12  17)  (0  3  6)  (7  II  10)  {15  18  19)  (1  8  14)   (2  5  20)   (9  13  16) 
To,=  (4  12  17)   (0  19  15)  (3  6  10)   {II  18  7)  (114  2)  (8  13  9)   (5  16  20) 
To,=  (4  12  17)  (o  3  II)  (6  18  19)   (7  15  io)(l  9  16)   (8  20  14)   (5  2  13) 
T,,— (4  12  17)   (07  6)  (3  IS  18)  (11  10  19)  (15  8)   (2  20  9)   (13  14  16) 

it  is  seen  that  the  vertices  of  no  one  of  the  triangles  2  8  16,  5  9  14,  1  20  14  are 
permuted  by  any  of  these  transformations  but  that  one  triangle  is  transformed 
wholly  into  another.  Since  the  squares  of  these  transformations  also  leave  F^.g 
invariant  there  are  twelve  transformations  of  type  I^,  of  period  three  leaving  Fg.g 
invariant  which  permute  the  vertices  of  a  given  one  of  the  fou»  triangles  whose 
vertices  are  in  R^,.,.  There  are,  therefore,  altogether  48  coUineations  of  type  If,  of 
period  three  which  leave  Y^,^  invariant. 

Summarizing,  we  have  in  the  total  group  leaving  F,,,3  invariant  9  clations,  8  type 
Ig's,  54  type  Ill's,  24  homologies,  72  type  II's,  48  t\'pe  Iq's  of  period  3,  and  the 
identity,  making  a  total  of  216  coUineations.  The  group  is  readily  identified  as 
the  Hessian  group  G^jo  discovered  by  C.  Jordan  in  187S.*  Here  the  9  points  of 
Fj,,  represent  the  9  points  of  inflection  of  each  cubic  of  the  pencil 

A.(a:i^+  x^^-\-x^')-\-ixx,x.,x,.=o 

which  is  invariant  under  every  collineation  of  the  group.  To  verify 
that  every  point  of  F9,3  lies  on  every  cubic  of  the  pencil  it  is  only  necessary 
to  notice  that  every  point  of  Y^,^  has  one  and  but  one  coordinate  zero  and 
z^^(r)'=l.      Every  subgroup  of  Goi^  leaves  F^jg  invariant  and  every  group  in 


*  Crelle,  Vol.  84,   (1878)  pp.  89-215.     The  gfroup  is  defined   by   leaving    invariant  a   pencil 
of  cubics  AF-f-^iH^O  where  F  is  a  ternary  cubic  and  H  its  Hessian  covariant. 


COLLINEATION    GrOUPS   OF    PG(2,2-).  27 

PG(2,2-)  (except  the  Goic  itself)  which  leaves  F^,^  invariant  must  be  a  sub- 
group of  the  G216.* 

(e)  T—^ oH 

^    ==F9,s'-  Consider  a  point   P  of  ¥^,/  and  the  three  lines  l^,l...L  of 

Fg.s'  which  pass  through  P.  On  1-^,1. 2,1,  are  7  points  of  Fg.^'  and  there  are,  there- 
fore, two  points  P'  and  P"  of  F,„/  not  on  l^jL,  or  I.,.  There  are  two  possibilities 
only — P'  and  P"  are  or  are  not  collinear  with  P..  If  P'  and  P"  are  collinear  with 
P  the  configuration  is  F^,,  except  that  3  lines  are  omitted.  If  P'  and  P''  are 
not  collinear  with  P  there  must  be  two  lines  through  P  (namely  PP'  and  PP") 
which  contain  but  two  points  of  F^,,'  each.  Since  P  can  be  transformed  into  any 
other  point  of  F,,,./  this  would  necessitate  the  existence  of  a  configuration  of  9 
points  arranged  two  points  to  the  line  which  contradicts  the  condition  that  when 
the  number  of  points  exceeds  6  there  must  be  at  least  3  to  the  line.  Hence  there 
is  but  one  possible  arrangement  and  that  is  as  the  9  points  and  9  of  the  lines  of 
Fgjs.  Since  Fg,^  includes  all  lines  joining  its  points  it  follows  that  every  trans- 
formation which  permutes  the  9  points  and  9  of  the  lines  of  Fg,3  also  permutes 
4mong  themselves  the  other  3  lines  of  Fg,3.  F^,/  is,  therefore,  the  subgroup  G-4  of 
order  54  of  Fq,^  which  leaves  invariant  a  simple  3-line  composed  of  3  lines  of  Fg,, 
so  chosen  that  no  two  of  them  meet  in  a  point  of  Fg,.. 

I  3      8  j  ==F8i3-  I>et  any  line  of  F^,.  be  chosen  as  the  line  x^=o.     Since 

but  6  lines  of  Fg.g  meet  ^^3=0  in  points  of  F8,3  there  is  one  and 
cRit  one  line  of  F^,,  which  meets  x.,=o  in  a  point  of  Rs,s-  Let  this  be  chosen  as 
the  line  x.^^o  and  let  a-^-=o  be  the  line  joining  the  two  points  on  x^=o  and  x.2=o 
(other  than  theix  point  of  intersection)  which  belong  to  R^.^.  The  points  3,  7, 
19  of  ^3=0  and  6^  11^  15  of  x.2=o  are  then  points  of  F^,.^.  Through  each  of  the 
points  3^  7j  19  pass  3  lines  of  Fs,3  which  contain  7  points  of  F^,^.  Since  the  one 
other  point  determines  but  one  line  with  a  given  point  it  follows  that  through  each 
of  the  points  3^  7^  19  passes  one  and  but  one  line  which  contains  no  other  point  of 
Fg,..  These  3  lines  must  meet  X2^=o  in  a  point  of  K^,^  and  hence  all  pass  through 
the  point  12  the  intersection  of  x.^^O  and  x^=o.  Any  Hpe  through  4  (1,0,0)  other 
chan  X2=0  and  ^^3^0  intersects  these  three  lines  in  points  of  Rs,3  and  hence  can 
contain  no  points  of  Fg,3  except  as  points  of  intersection  with  x^^o.  Therefore  the 
other  two  points  of  Fs,3  lie  on  Xi=o.  But  we  know  that  the  9  points  other  than 
vertices  on  the  sides  of  any  triangle  in  PG(2,2-)  are  collinear  by  threes  on  12  lines 
of  which  4  pass  through  each  point.  Accordingly,  if  any  one  of  the  points  0,  10, 
18  on  x^=-o  be  omitted  there  remain  8  points  collinear  by  threes  on  8  lines.     Hence 

*  The  group  G216  was  studied  at  length  by  Maschke,  Mat/i.  Anna/en,  Vo\.  33  (1890),  pp. 
324-330,  and  the  geometric  properties  of  the  group  and  its  subgroups  by  Newson,  Kansas  Uni- 
'versity  Quarterly,  Vol.  II,  No.  6  (Apr.  1901),  pp.  13-22. 


28  U.  G.  Mitchell:  Geometry  and 

Fs,3  must  be  F,,,.-;  with  one  point  and  the  4  lines  throvi}j:h  that  point  omitted  and  its 
group  is  the  subgroup  of  F,,..  G04  of  order  24  which  leaves  invariant  a  single  point. 

(e)    r^ w-\ 

^1  .=F-,..  Let  /  be  anv  line  of  F-,.,  and  P,  Q,  R  the  three  points  on 

3       /    I  '  •' 


/  belonging  to  F-,.,.  Then  there  are  but  four  other  points  A,B,C,D  of  F-,, 
not  on  /  and  these  four  points  must  lie  two  b\'  two  on  two  lines  through 
each  of  the  points  of  the  complete  quadrilateral  of  the  other  four  points  A,B,C,D. 
Let  A,B,C,  be  taken  (See  Fig.  1,  p.  4)  as  the  vertices  (/, 0,0)  ,(0,0,7)  (o,/,o) 
respectively  of  the  triangle  of  reference  and  D  as  the  point  {1,1,1).  This  deter- 
mines the  coordinates  of  P,Q,R  as  {T,o,i),{o,r,i),  {1,1,0)  respectively.  Since 
these  coordinates  are  in  the  GF(2)  F-,.  coincides  with  PG(2,2)  and  we  can  de- 
termine the  number  of  collineations  of  each  type  by  substituting  n^=l  in  the 
formulae  given  on  p.  10,  noting  that  type  I,  of  PG(2,2)is  type  I.  of  PG(2,2-). 
This  gives  56  collineations  of  type  I3  (type  I^  in  PG(2,2)  ),  48  of  type  Ip  of 
period  7,  42  of  type  III,  and  21  elations,  which,  together  with  the  identical  trans- 
formation make  a  group  G^^^  o^  order  108.  Every  collineation  in  the  group  is 
of  determinant  unity  and  since  it  is  of  degree  7  and  order  168  it  is  recognized  as 
the  smiple  group  G],.,s  first  derived  by  Klein  by  the  consideration  of  the  trans- 
formation of  the  seventh  order  of  elliptic  functions.* 

Every  group  which  leaves  F-,,  invariant  must  be  a  subgroup  of  Gj,,,s  and  every 
subgroup  of  the  G^,-^  leaves  F^,.,  invariant^ 


(0  r^^—r-.  (g) 


2     15  I        «'''  I  2     12 

(i) 


(h) 


=F     '•  I   6      3  j^p     , 

I   2      fl   I         «•' 


^      2  |_p 
2      6   I         '•'- 


Since  the  six  points  of  F^,.^  are  no  three  coUinear  we  may  choose  any  three  of 
them  for  the  vertices  4  (7,0,0),  12  (0,0,7),  17  (0,7,0)  of  the  triangle  of  reference 
and  any  point  not  collinear  with  any  two  of  these,  say  {i.i.i)  may  be  taken  as  the 
fourth  point.  Since  in  PG(2,2=)  the  choice  of  four  points  no  three  of  which  are  col- 
linear determines  uniquely  (Cf.  Corollaries  4  and  5  of  Theorem  4)  the  other  two 
points  not  collinear  with  any  two  of  them,  the  fifth  and  sixth  points  are  neces- 
sarily 13  {i,i,i)   and  20   {i,ij).     Six  points,  no  three  of  which  are  collinear,  de- 

*  Math.  Annalen,  Vol.  14  (1878),  p.  438.  Jordan,  in  determining-  the  finite  ternary  groups 
missed  both  this  group  and  the  simple  G36O.  The  G168.  is  discussed  at  some  length  by  Burnside, 
Theory  of  Groups,  pp.  208-209  and  302-305,  and  in  Klein-Fricke's  Modulfunctionen. 

*  For  a  list  of  all  groups  whose  degree  does  not  exceed  8,  see  Miller, ^w^r.  Journal  of  Math. 
Vol. 21(1899),  p.  326.  The  types  of  substitutions  and  of  subgroups  of  the  G168  are  given  by  GoR- 
DAN,  Math.  Annalen,  Vol.  25,  (1885),  p.  462. 


COLLINEATFON    GrOUPS   OF    PG(2,2-).  29 

termine  15  distinct  lines.  Hence  every  collineation  which  leaves  invariant  the 
six-point  also  leaves  invariant  the  associated  fifteen-line.  From  this  is  follows 
that  the  groups  leaving  F^,,',  ^ (^,,-2" ,  F^,., '"  ,  invariant  must  either  be  the  group 
leaving  F,,,o  invariant  or  subgroups  of  it. 

An  elation  E  leaving  Fr,o  invariant  can  not  have  more  than  two  points  of  Fg,, 
on  its  axis,  and  since  an  elation  interchanges  by  pairs  all  points  not  on  its  axis  V. 
must  interchange  by  pairs  at  least  four  points  of  F,;,,.  Since  any  four  points  no 
three  of  which  are  collinear  can  be  transformed  into  any  four  such  points  by  a 
collineation  there  exist  in  PG(2,2-)  collineations  interchanging  by  pairs  any  four 
points  of  F,;,2.  All  such  collineations  are  elations  because  no  other  transforma- 
tions in  PG(2,2-)  are  of  period  two.  Moreover,  such  an  elation  E  leaves  inva- 
riant each  diagonal  point  of  the  complete  quadrangle  of  the  four  points  chosen 
and  therefore  the  other  two  points  on  the  diagonal  line  which  are  not  diagonal 
points.  These  last  two  points  must  be  the  other  two  points  of  F,.,o  (Cf.  proof  of 
Cor.  5,  Theorem  4).  Since  four  points  no  three  of  which  arc  collinear  can  be 
interchanged  by  pairs  in  three  different  ways  it  follows  that  there  are  three  ela- 
tions leaving  Fr,,o  invariant  for  each  distinct  quadrangle  that  can  be  chosen  from 
Fn,o.     There  are  then  3(0 -5 -4 -3/4  !)==-!.")  elations  which  leave  F,;,.^  invariant. 

Since  any  four  points  no  three  of  which  are  collinear  can  be  transformed  into 
any  four  such  points  by  a  collineation  there  exist  in  PG(2,2-)  collineations  per- 
muting any  four  points  of  F,,,„  in  any  given  cyclic  order.  Such  a  transformation 
must  be  of  period  four  and  therefore  of  type  III.  A  collineation  T  of  type  III 
which  permutes  in  cj'clic  order  any  four  points  of  F,.,,o  must  leave  invariant  one 
of  the  diagonal  points  of  the  complete  quadrangle  of  the  four  points  and  inter- 
change the  other  two  diagonal  points.  Since  any  two  points  of  Fo,o  lie  on  the 
diagonal  line  of  the  complete  quadrangle  of  the  other  four  points  but  are  not 
diagonal  points  it  follows  that  if  T  permutes  in  cyclic  order  four  points  of  F,.,,, 
it  interchanges  the  other  two  points  of  F(.,o.  Since  any  four  points,  no  three  of  which 
are  collinear,  can  be  permuted  in  six  different  cyclic  orders  it  follows  that  there 
are  6  (6 -5 -4 -.3/4!)  =90  collineations  of  type  III  which  leave  F,;,o  invariant. 

Since  a  collineation  of  type  I,  of  period  5  leaves  invariant  one  real  point  if  it 
leave  Fg,o  invariant  its  center  must  be  a  point  of  F,,,.,.  The  other  points  of 
}'(.,, 2  form  a  conic  of  which  the  sixth  point  (the  center)  is  the  outside  point.  If 
4  (/,o,o)  be  taken  as  the  center  and  the  other  five  points  permuted  in  the  order 
(1  13  20  12  17)  the  transformation  is  found  to  be 

T:     f)X.,'^rx.;,~\-x^ 
px.,'^x., 

of  type  Ij  and  period  ').  It  was  shown  in  §  3  that  there  are  six  different  pairs  of 
imaginary  points  on  the  axis  of  T  determining  six  transformations  of  type  Ij  with 
the  same  center  and  axis  and  no  one  a  power  of  another.  These  correspond  to  the 
six  different  cyclic  orders  in  which  five  points  of  the  conic  can  be  permuted  and 
there  are,   therefore,   C   independent  transformations  of  period  5    (24  altogether) 


30  1j.  G.  Mitchell:  Geometry  and 

leaving  Fe,o  invariant  and  having  the  point  4  for  center.  Hence  there  are  alto- 
gether 6  •24=14-1  collineations  of  type  Ij  leaving  Fq,2  invariant. 

Since  a  transformation  of  type  I^  permutes  in  cyclic  order  any  set  of  three  points 
so  related  to  the  vertices  of  its  invariant  triangle  that  the  six  points  are  no  three 
collinear,  any  coUineation  of  type  lo  which  has  three  points  of  F,.,2  for  vertices 
of  its  invariant  triangle  must  leave  F^,^  invariant.  Twenty  distinct  triangles  can 
be  chosen  from  the  six  points  of  F,.,o  and  hence  40  collineations  of  type  I3  leave 
Fg,o  invariant  in  this  way.  We  know,  however,  that  each  such  coUineation  per- 
mutes the  vertices  of  two  triangles  not  in  Fq,.^  either  of  which  may  be  taken  as 
the  invariant  triangle  of  a  transformation  of  type  I3  which  permutes  the  vertices  of 
the  two  triangles  in  Fp,^.  For  each  pair  of  triangles  that  can  be  selected  in  Fc,2 
there  are  then  four  transformations  of  type  I3  having  invariant  triangle  not  in 
Fe,2  which  leave  Fj,,2  invariant.  Since  ten  pairs  of  triangles  can  be  chosen  in  Fq,^ 
there  are  40  collineations  of  type  I3  which  leave  F^jo  invariant  in  this  way. 

It  has  been  shown  that  the  group  which  leaves  Fg,o  invariant  must  contain 
as  many  as  360  collineations.  Since  the  group  can  be  represented  as  a  substitu- 
tion group  on  fi  symbols  it  must  therefore  be  either  the  alternating  or  symmetric 
group  of  degree  six.  But  since  there  is  no  coUineation  which  holds  four  of  the 
[joints  of  Fg,o  each  fixed  and  interchanges  the  other  two  the  group  can  not  be  the 
symmetric  group.  The  group  which  leaves  Fgjo  invariant  is  therefore  the  alternat- 
ing group  on  six  symbols,  shown  by  Wiman*  to  be  identical  abstractly  with  the 
the  finite  ternary  group  Go^.o  first  set  up  by  H.  Valentiner.f 

The  group  G3,;,,  is  here  characterized  not  only  as  a  group  on  six  points  but 
since  any  one  of  the  points  can  be  taken  as  the  outside  point  of  the  conic  deter- 
mined by  the  other  five  points  as  a  group  leaving  invariant  a  system  of  six  conies. 
It  is  clear  that  every  subgroup  of  G-t,;,,  leaves  F,,,^  invariant  and  every  group  in 
PG(2,2-)  which  leaves  Fy,o  invariant  must  be  a  subgroup  of  G^^.o. 


(J) 


5      4 
2     10 


=F,,2; 


(k) 


o 


D 


T.., 


Since  the  five  points  of  F-,2  are  no  three  collinear  they  form  a  conic  and  the 
group  which  leaves  F_r,,2  invariant  is  therefore  simply  isomorphic  with  the  group 
cf  all  transformations  of  points  on  a  line.  The  group  accordingly  contains  15 
elations,  20  type  I.,'s  and  24  type  I/s  of  period  5  (Cf.  §3).  Since  every  trans- 
iormation  which  leaves  a  conic  invariant  must  leave  its  outside  point  invariant 
this  group  is  recognized  as  the  subgroup  of  the  G.j,.,,,  which  leaves  a  single  point 
fixed.  It  is  liere  represented  'Joth  as  the  group  which  leaves  invariant  a  conic  and 
the  alternating  group  on  five  svmKols( the  five  points  of  Fr,,o). The  group  which 
leaves  F.,^'  invariant  must  be  the  subgroup  of  the  G^y  which  leaves  the  10  lines  of 

*  Mai/i.  Afinalcn,  Vol.  47,  (1896),  p.  531. 

t  /(/Wa  Wr.  (5)  5  (1889),  p.  64.  See  Eftcj.  J.  M»t//.,  IViss.,  Vol.  I,  p.  529.  In  deter- 
mining the  finite  ternary  groups,  Valentiner,  who  was  apparently  unaware  of  the  previous  work,  of 
Klein  and  Jordan,  ini.sscdthe  Gj6u. 


COLLINEATION    GrOUPS   OF    PG(2,2-).  3I 

Fq.o  invariant  in  two  systems  of  five  lines  each.     It  is  therefore  a  group  G^^,  of 
order  10. 


^4'2.  I     2  4     j  4'2    • 


The  configuration  F^,,  is  a  complete  quadrangle  and  since  four  points  can  be 
permuted  among  themselves  in  all  ways  by  transformations  in  the  plane  the  group 
must  be  the  symmetric  group  Go^  of  all  transformations  on  the  four  points.  The 
configuration  F^,.'  is  a  simple  quadrangle  and  hence  its  group  is  the  subgroup  G^ 
of  the  G24. 

^"^"^"2l      ^  .  . 

12      3     =1^3,2.  The  configuration  F3,o   is  the  triangle  and   hence  every 

transformation  leaving  it  invariant  must  do  so  in  some  one  of  the  following  three 
ways:  (1)  Leave  the  three  vertices  each  fixed;  (2)  leave  one  vertex  fixed  and 
interchange  the  other  two;  (3)  permute  the  three  vertices  in  cyclic  order. 

These  may  be  written  down  at  once  as  follows: 

Under  (1)  theie  are  2  type  Ig's  and  6  homologies;  under  (2)  there  are  18 
type  II's  and  9  elalions;  under  (3)  there  are  6  type  L/s  and  12  type  I^'s  of  period 
3.     Including  the  identity,  then,  there  are  54  collineations  in  the  group. 

§  8.  Subgroups  of  the  Group  Gog^;,,  Which  Leaves  a  Line  Invariant. 

All  groups  which  leave  invariant  a  set  of  collinear  points  must  also  leave  in- 
variant the  line  /  which  contains  the  points.  Since  any  four  lines  no  three  of  which 
are  concurrent  can  be  transformed  into  any  four  such  lines  by  a  projective  trans- 
formation in  PG(2,p")  the  order  of  the  group  leaving  a  line  fixed  is 
N=(/)-"-f/)")  (/)-'")/>'-'" — 2/>" -fl)  =/.'"(/)-"— 1)  (/)"—l) 

For  PG(2,22)  this  gives  N=2S80  and  accordingly  the  group  will  be  desig- 
nated as  Go„,so'  Since  any  line  in  the  plane  can  be  transformed  into  any  other  line 
m  the  plane  by  a  coUineation  within  the  Goo48o  it  follows  that  G^oiso  (-ontains  21 
conjugate  groups  Goggo- 

Subgroups  of  G-ggo  i^hich  leai  c  a  point  not  on  I  invariant.  In  determining  the 
subgroups  of  G0S80  we  shall  first  determine  all  subgroups  which  leave  invariant  at 
least  one  fioint  not  on  /  and  then  all  subgroups  which  leave  invariant  no  point  not  on 
/,  Taking  /  as  the  line  x.^=o  (or  the  line  at  infinity)  every  coUineation  in  the 
0,84.  is  of  the  form 

_       x'=a^x-{-a.,y-\-a 
^'-     y'=b,x+b,y^-b 
where  x   and  y   are  nonhomogeneous  point  coordinates.      Selecting  the   point   not 
on  /  as  tie  origin  every  coUineation   in  the  Gogso  which   leaves   it   invariant   is  of 

the  form 

„,  _  x'^a^x-\^a.,y 
'  "  y'=bj^x-]-b...y 
But  it  has  been  seen  in  §  3  that  in  homogeneous  coordinates  the  group  of  all  trans- 


32  U.  G.  Mitchell:  Geometry  and 

formations  of  the  form  T^,  is  the  group  G^;^,  of  all  transformations  of  points  on 
a  line.  Since  G^,,  is  the  alternating  group  on  five  symbols  it  contains  the  follow- 
mg  subgroups:* 

I.  Subgroup  leaving  all  points  of  the  line  fixed. 

1  self-conjugate  G, — the  identity.     T,  :  Ar'=rt,A-,  y'^=a^^y. 

II.  Subgroups  lca\"ing  invariant  two  points  of  the  line. 

a.  Those  leaving  each  point  of  the  pair  fixed. 

10  groups   G3   each   conjugate  to   the  G^   of   transformations  of   the   form 
T.r.  x'=a^x,  y'=lj.j\  rt,,  h..  in  the  GF(2-), 

b.  Those  leaving  the  pair  invariant. 

10  groups  G,;  each  conjugate  to  the  Gg  of  transformations  of  the  form  T^ 
subject  to  the  restriction  that  cither  a^=b._,=o  or  a.,=b^^o. 
10  groups  Go  each  conjugate  to  the  subgroup  of  G^  for  which  the  coeffi- 
cients are  in  the  GF(2). 

III.  Subgroups  leaving  one  point  of  the  line  invariant. 

15   groups  Go  each  conjugate   to   the  Go   of   transformations   of   the   form 
T.^:  x'=a^x-^any,  y'=  a^y,  where  a^,an,'Avt  in   the  GF(2-). 
5  groups  G4  each  conjugate  to  the  G4  of  transformations  of  the  form  T3 
where  a^,  a.,  are  in  the  GF(2-). 

5  groups  Gjo  each  conjugate  to  che  Gjo  of  all  transformations  of  tlie  form 
T^ :  x'=a^x-\  a.,y,  >''=^2  y,  where  a^,a.,bo^r(;  in  the  GF(2-'). 

IV.  Subgroups  leaving  invariant  a  pair  of  imaginary  points  on  the  line. 

a.  Those  leaving  each  point  of  the  pair  fixed. 

6  groups  G-  each  conjugate  to  the  G-  of  all  transformations  of  the  form 
T^  subject  to  the  condition  that 

b.  Those  leaving  the  pair  invariant. 

6  groups  Gjo  each  conjugate  to  the  G,„  of  all  transformations  of  the  form 
To  subject  to  the  condition  that 

(rii--\-in.r-\-rb,-^b.r-}-a,(i.,-\-r^^b^-^ra.,b^^a.,b.,  -\-rb,b.,)  {a,-\-b^-\-in.,)^o 

If  T„  be  taken  as  a  transformation  in  nonhomogeneous  coordinates  we  have  a 
group  G,;^,  simply  isomorphic  with  G,.„''  or  a  group  G.^,,  triply  isomorphic  with 
G,;,,^  according  as  the  determinant  of  the  group  of  transformations  of  the  form  T, 
is  unity  or  unrestricted  within  the  GF(2-).  Corresponding  to  the  above  groups  on 
the  Ime  there  are  then  the  following  groups  in  the  plane  which  leave  invariant  the 
line  /  and  point  {0,0). 

I.  Subgroups  leaving  all  points  of  /  fixed. 

3.  Of  determinant  unity,  1  self-con iugate  Gj. 

2.  Of  determinant  not  restricted,  1  self -con  jugate  Gy. 

II.  Subgroups  leaving  invariant  a  pair  of  points  on  /. 
1.  Those  leaving  each  point  of  the  paii   fixed. 

a.  Of  determinant  unity,  10  conjugate  G..  each  leaving  a  triangle  invariant. 

b.  Of  determinant  not  restricted,  10  conjugate  Gy  each  leaving  a  triangle  in- 
variant. 


*  All  groups  of  degree  less  than  6  were  obtained   bv  Se 


COLLINEATION    GrOUPS    OF    PG(2,2-).  3;j 

2.  Those  leaving  the  pair  of  points  invariant. 

a.  Of  determinant  unity, 

10  conjugate  G,,  each  leaving  a  triangle  invariant, 

b.  Of  determinant  unrestricted, 

10  conjugate  G^g  each  leaving  a  triangle  invariant. 
III..  Subgroups  leaving  one  point  of  /  fixed. 
1.  Of  determinant  unity, 

15  conjugate  G2  each  leaving  invariant  a  point  of  lines. 
5  conjugate  G4  each  leaving  invariant  a  point  of  lines. 
5  conjugate  G,,,  each  leaving  invariant  the  line  /,  the  jr-axis  and  the  origin 
IV.     Subgroups  leaving  invariant  a  pair  of  imaginary  points  on  /. 

1.  Subgroups  leaving  each  point  of  the  pair  fixed. 
6  conjugate  Gr,  of  determinant  unity. 

6  conjugate  Gj^  of  determinant  not  restricted. 

2.  Subgroups  leaving  the  pair  invariant. 
6  conjugate  Gjo  of  determinant  unity. 

6  conjugate  G30  of  determinant  not  restricted. 
Subgroups  of  G  .ssu  ii~'hich  leave  no  point  not  on  I  invariant.  In  the  discussion 
which  follows  the  term  translation  will  be  used  to  Indicate  an  elation  having  the 
line  /  for  axis  and  the  term  elation  will  be  used  only  for  an  elation  whose  axis  Is 
not  /.  In  determining  the  subgroups  of  Gogso  which  leave  invariant  no  point  not 
on  /  we  shall  first  determine  all  such  subgroups  containing  no  translations  and 
then  all  such  subgroups  containing  translations. 

Let  Gn  be  a  subgroup  of  Goggr,  which  leaves  no  point  not  on  /  invariant  and 
contains  no  translation.  If  Gn  contain  an  homology  H  having  /  for  axis  and  A 
for  center,  G„  must  contain  at  least  one  transform  H'  of  H  having  some  other 
point  than  A  for  center.  One  of  the  products  H'H  or  H'H-  is  of  determinant 
unity  and  leaves  all  points  on  /  fixed.  It  is  therefore  a  translation.  Hence  Gn  can 
contain  no  coUineation  other  than  the  identity  leaving  all  points  of  /  fixed  and, 
consequently,  every  such  group  must  be  simply  isomorphic  with  the  Ggo- 

We  have  seen  (ante  p.  30)  that  there  is  a  group  G,,f,  leaving  invariant  a  point 
conic  and  its  outside  point.  By  duality  there  is  a  Gbo  leaving  invariant  a  line 
conic  and  its  outside  line.  Since  the  line  /  is  the  outside  line  of  48  different  line 
conies  there  are  48  such  groups  Geo  which  leave  /  invariant. 

Every  transformation  of  the  form 

x'=  X  -\-  a  ■ 

is  a  translation  and  the  group  of  all  transformations  of  the  form  E  where  a  and 
b  are  marks  of  the  GF(2-)  is  a  G^g  leaving  every  point  on  /  fixed.  Unless 
a=b==o  E  is  of  period  2  and  hence  G^g  contains  15  cyclic  subgroups  of  order  two. 
If  a^o  and  b  be  allowed  to  take  on  all  values  in  the  GF(2-)  or  if  b=o  and  n 
be  in  the  GF(2-)  a  group  of  order  4  is  obtained,  consisting  of  all  translations 
leaving  fixed  all  lines  through  a  given  point  P  on  /.  Such  a  group  will  be  desig- 
nated as  a  G4(P).  If  a  be  allowed  to  take  the  value  0  and  but  one  other  value 
and  b  be  restricted  in  the  same  way  a  group  of  order  4  is  obtained  containing  be- 
sides the  identity  3  translations  no  two  of  which  have  the  same  center.  Since  such 
a  group  leaves  invariant  a  complete  quadrangle  of  which  the  centers  of  the  three 
elations  are  the  diagonal  points,  it  will  be  designated  as  a  0^(0).     If  a  be  in  the 


34 


U.  G.   Mitchell:   Geometry  and 


GF(2-)  and  b  in  the  GF(2)  a  group  G^  is  obtained  leaving  invariant  a  point  P 
on  /  and  interchanging  the  four  lines  other  than  /  through  P  by  pairs  in  a  given 
manner. 

The  Gic  is  an  Abelian  (or  commutative)  group  since  if  Ei  and  Ej  be  any  two 
clations  in  G^e  EjEi=EiEj.  Consequently  EjEiEf^=EiEjEj-^=Ei,  and  every 
.nibgroup  of  Gjg  is  self-conjugate  within  the  Gjc-  Also  every  two  translations  and 
their  product  form  (with  the  identity)  a  group  G4,  for  if  EiEj=Eit, 
E,=EkEj=EjEk  and  Ej=EkEi=EiEk.  Hence  G^r,  contains  l") •14/0=3')  sub- 
groups G4.* 

We  shall  next  determine  the  subgroups  of  G^ssn  which  are  such  that  c\ery  col- 
lineation  in  the  group  either  lea\es  invariant  a  point  not  on  /  or  is  the  product  of 
such  a  collineation  and  a  translation  in  the  group. 


*  It    may   be   of  interest   to  note  that  the  Gi6can  be  represented  as  a  three-space  PG(3,2)  by 
letting  the  G?.  G4,  Gs,  correspond  to  the  points,  lines  and  planes,  respectively,  of  the  three-space. 


The  three -space   S3  has 

The  Group  G16  has 

15   points,    35  lines,  15  planes, 

15  subgroups  G2,   35G4,  15  G?^, 

arranged 

arranged 

3  pf)ints  on  each  line, 

3  G2  in  each  G4, 

7  points  on  each  plane , 

7  G2  in  each  Gs, 

7  lines  through  each  point. 

7  G4  containing  each  G2, 

7  lines  in  each  j)lane, 

7  G4  contained  in  each  Gs, 

3  planes  through  each  line. 

3  Gs  containing  each  G4, 

7  planes  through   each  point. 

7  Gs  containing  each  G2. 

If  the  three-space  S?  be  represented  by  the  notation  for  a  configuration  (Cf.  MooRE,  American 
Journal  oj  Mathematics,  Vol.  18,  pp.  264-303;  Veblf.n  and  You.vg,  Projeaive  Geometry.  Vol.  I,  p 
38),  the  same  table  exhibits  the  structure  of  the  group  G|6. 


G„ 

G2 

G4    Gs 

\ 

S3 

So 

Si       b2 

G2—  So 

15 

7     7 

G4  —   Si 

3 

35     3 

Gs  —    S2 

7 

7  15 

In  the  table.  So  is  a  point.  Si  a  line,  S2  a  plane,  and    the    interpretation    is    obvious    from    the 
parallelism  given  above. 


COLLIXEATION    GrOUPS    OF    PG(2,2-).  35 

The  translations  in  any  subgroup  of  Gogso  form  a  self-conjugate  subgroup  Gk. 
If  any  group  G„  has  a  system  of  transitivity  S  which  is  also  a  system  of  transitivity 
of  its  self-conjugate  subgroup  Gk  of  translations  then  every  collineation  in  Gn  is 
either  a  collineation  leaving  a  pomt  O  of  S  invariant  or  the  product  of  such  a  col- 
lineation and  a  translation ;  for,  let  O  be  taken  as  the  origin  and  let  T  be  any 
collineation  in  Gn-  If  T  displaces  O  it  changes  O  to  some  point  A  in  S.  But  since 
S  is  a  system  of  transitivity  for  Gk  there  is  in  Gq  a  translation  T^  changing  A  to  O. 
Hence  TiT=Tn  a  collineation  in  Gn  leaving  O  invariant.  From  T^T=T2 
we  have  T=TiT2.  Every  such  group  Gn  can  be  obtained  then  by  extending 
the  groups  leaving  a  point  fixed  by  means  of  translations.  In  determining  the 
groups  below,  S  will  be  used  to  indicate  the  system  of  transitivity  common  to  the 
group  obtained  and  the  extending  group  of  translations.  Ej,  E,  and  E  will  be 
used  to  indicate  the  forms  of  translations  as  follows: 

J?  =^  x'=x-\-a  Y  ^^  x'=x  ■n'=  x^=x-^a 

The  product  of  To=  ^'=''.^"+«2>'       and  E  is  of  the  form 
y'=b^x-\-b2y 

ry  Y ^_x^ =a.^x-\-a^y -\-a.ji  -{-dob 
"         y'^b.^x-\-b^y-\-b^a-\-b2b 

It  should,  therefore,  be  observed  that  in  extending  a  group  of  coUineations  of 
the  form  Tq  by  E^^,  £3  or  E  the  range  of  values  which  can  be  assumed  by  the 
additive  constants  in  the  product  is  dependent  upon  the  coefficients  of  Tq  as  well 
as  those  of  E^  Eo  or  E.  In  the  work  below  S  will  be  used  to  indicate  a  system  of 
transitivity  of  the  extended  group  G„  which  is  also  a  system  of  transitivity  of  the 
extending  group  Gk  of  translations.  The  groups  obtained  by  such  extensions  are 
as  follows: 

I.  Subgroups  leaving  each  point  of  I  fixed. 

1.  Extensions  of  Gi  by  Ei,E2,E  give  groups  of  translations  only. 

2.  Extensions  of  the  G3  of  the  form  T^:  x'=  a^x,y^=a^y,  «,  in  GF(2-). 

a.  By  E;^  gives  a  G12   leaving  invariant  the   Ar-axis.     The  points   on   the 
AT-axis  form  the  system  S, 

b.  By  Elo  gives  a  similar  G^o  leaving  invariant  the  >'-axis. 

c.  By  E  gives  a  G^g  leaving  invariant  no  point  not  on  /  and  no  line  but  /. 
The  system  S  includes  all  points  not  on  /. 

II.  Subgroups  leaving  a  pair  of  points  on  I  invariant. 

■    1.  Subgroups  leaving  each  point  of  the  pair  fixed.     (10  of  each). 

A.     Those    of    determinant    unity-      Extensions   of    G.    of    the    form   T., : 
x'=a^x,  y^=b2y,  Cy  and  b^  in  the  GF(2-). 
a.  By  El  gives  a  Gjo  leaving  the  jc-axis  invariant. 
The  system  S  includes  all  points  on  the  Ar-axis.. 
.  b.  By  E,  gives  a  similar  Q,.,  leaving  the  >-axis  invariant, 
c.   By  E  gives  a  G^^  leaving  invariant  no  point  not  on  /  and  no  line  hut  /. 
The  system  S  includes  all  points  not  on  /. 


36  L'-   G.    Mitchell:   Geometry  and 

B.  Those  of  determinant  not  restricted.     Extensions  of  the  Gg  of  the  form 
T,.     (10  of  each). 

a.  By  Ej  gives  a  G^^  leaving  the  Ar-axis  invariant. 
The  system  S  includes  all  points  on  the  Ar-axis. 

b.  By  E,  gives  a  similar  Gg,,  leaving  the  >'-axis  invariant. 

c.  By  E  gives  a  G^^^  leaving  invariant  no  point  on  /  and  no  line  but  /. 
The  system  S  includes  all  points  not  on  /. 

2.   Subgroups   leaving  the   two   points  invariant   as  a   pair. 

A.  Those  of  determinant  unity.      (10  of  each). 

Extensions    of    the    G.    of    T,,    with    the    form    Oy=b.^=o  or  a.^^-=l^=o, 
a-^.a^,b^,b..,  being  In  the  GF(2). 

a.  By  Ej  extends  by  E  also  giving 

For  a  in  GF(2)  a  Gg  leaving  invariant  a  PG(2,2). 
The  system  S  includes  four  points  not  on  /  no  three  of  which  are  coUinear. 
For  a  in  the  GF(2-)  a  G^o  leaving  invariant  no  point  not  on  /  and  no 
line  but  /.    The  system  S  includes  all  points  not  on  /. 

b.  By  Eo  gives  same  as  by  E^  or  E. 

Extension  of  the  G,.  of  the  form  T^,     with  aj^=b.^^o.  or  a^=b^^o  and 
a^,b^,a^,b^  in  the  GFi2'-). 

By  Ej  or  Eo  extends  by  E  giving  a  G;,^  leaving  invariant  no  point  not 
on  /  and  no  line  but  /.     The  system  S  includes  all  points  not  on  /. 

B.  Those  of  determinant  not  restricted.     (10  of  each) 

Extension  of  the  G^,  by  Ej  or  E^  extends  by  E  with  a  and  b  unrestricted 
giving  a  G^^g  leaving  invariant  no  point  not  on  /  and  no  line  but  /. 
The  system  S  includes  all  points  not  on  /. 

III.  Subgroups  leaving  one  point  on  I  fixed. 
1.  Those  of  determinant  unity. 

A.  Extensions  of  G,  of  form  l^^:x'=  a.^x-\-n.,y,  y'=^a,y,  a^,a^  in  GF(2). 

a.  By  E^  with  a  in  the  GF(2)   gives  a  G^  leaving  invariant  the  A--axis. 

( 1-3  such  groups ) .  The  system  S  is  the  points  on  the  A--axis  having  coor- 
dinates in  the  GF(2). 

b.  By  Ej  with  a  in  the  GF(2-)  gives  a  G^  leaving  the  A-axis  invariant. 
The  system  S  includes  all  points  on  the  Ar-axis. 

c.  By  Eo  with  b  in  the  GF(2)  extends  by  E  with  a  and  b  in  the  GF(2) 
giving  a  Gg  leaving  invariant  a  PG(2.2)  which  is  also  the  system  S. 

d.  By  E_.  with  b  in  the  GF(2-)   gives  a  G,o  leaving  invariant  no  point 
not  on  /  and  no  line  but  /.    The  system  S  includes  all  points  not  on  /. 

B.  Extensions  of  G^  of  form  T^  with  a^,  a.,  in  GF(2-).     (5  of  each), 
a-  By  Ej  gives  a  Gi,,  leaving  invariant  the  .v-axis. 

The  system  S  includes  all  points  on  the  ;f-axis. 
b.   By  Eo  extends  by  E  giving  G,.,^  leaving  invariant  no  point  not   on   / 
and  no  line  but  /.      The  system  S  includes  all  points  not  on  /. 

C.  Extension  of  Gj.  of  form  T.:    x'^a^x  -{-  a^y,  y'=b^y.  (.5  of  each). 
^-   ^y  El  gives  a  G^^  leaving  in\ariant  the  x-axis. 

The  system  S  includes  all  points  on  the  Ar-axis. 
b.   By  E,  extends  by  E  giving  a  G,,^  leaving  invariant  no  point  not  on  / 
and  no  line  but  /. 
The  system  S  includes  all  points  on  the  A--axis. 


CoLLiNEATioN  Groups  OF  PG(2;2-).  37 

2.  Those  of  determinant  not  restricted.    (5  of  each). 

A.  Extensions  of  G^o  of  form  T4. 

a.  By  El  gives  a  G^g  leaving  a  point  of  lines  invariant. 
The  system  S  includes  all  points  on  the  Ar-axis. 

b.  By  E,  extends  by  E  giving  a  G,,,^  leaving  invariant  no  point  not  on 
/  and  no  line  but  /.   The  system  S  includes  all  points  not  on  /. 

B.  Extensions  of  G..,;  of  form  T-. 

a.  By  Ej  gives  a  Gj44  leaving  the  x-axis  invariant. 
The  system  S  includes  all  points  on  the  ^--axis. 

b.  By  Eo  extends  by  E  giving  a  G--,.,  leaving  invariant  no  point  not  on 
/  and  no  line  but  /.    The  system  S  includes  all  points  not  on  /. 

IV.  Subgroups  leaving  invariant  a  pair  of  imaginary  points  on  I.     {6  of  each). 
In  each  case  the  system  S  includes  all  points  not  on  /  and  no  point  on  /  and  no 
line  other  than  /  is  invariant. 

1.  Subgroups  leaving  each  point  of  the  pair  fixed. 

A.  Of  determinant  unity. 

Extension  of  G.-   of   form  T„    (subject  to  quadratic  condition)    by   E^ 
or  E,  extends  by  E  giving  a  G<jo. 

B.  Of  determinant  not  restricted. 

Extension  of  Gi,-  of  form  Tn    (subject  to  quadratic  condition)    by  Ej 
or  Eo  extends  by  E  giving  a  Go^n- 

2.  Subgroups  leaving  the  points  invariant  as  a  pair. 

A.  Of  determinant  unity. 

Extension  of  Gj,,  of  form  T^  (subject  to  cubic  condition)   by  Ej  or  E_. 
extends  by  E  giving  a  G^qo- 

B.  Of  determinant  not  restricted. 

Extension  of  G.,,,  of  Form  T,,  (subject  to  cubic  condition)  by  E,  or  E^ 
extends  by  E  giving  a  G^go- 
The  extension  of  the  total   group   Ggo  of   the   form  T„  and   determinant   unity 
by  E^  or  E,  extends  by  E  giving  a  group  Ggco  oi  all  transformations  of  determi- 
nant unity  in  the  Goggo-     The  extension  of  the  group  Gigo  of  form  Tq  by  Ej  or 
E2  extends  by  E  giving  the  Gnggo  itself. 

There  remain  to  be  determined  the  subgroups  of  G._,g„„  which  leave  no  point 
not  on  /  invariant,  contain  translations,  and  are  such  that  the  self-conjugate  sub- 
group of  translations  has  no  system  of  transitivity  which  is  also  a  system  of  tran- 
sitivity for  all  other  transformations  in  the  group.  Since  the  group  G,„  of  trans- 
lations is  transitive  on  all  points  not  on  /  no  such  subgroup  can  contain  the  Gj,,, 

Let  Gn  be  such  a  subgroup  of  Goggo  containing  a  G,  as  the  largest  self-conju- 
gate subgroup  of  translations.  Selecting  S :  ;if'=x-|-/^  J''=.V  as  the  trans- 
lation in  the  G_.  and  T:  x'==a,x-\-h^y-\-c^,  y'==(i.,x-\^b._,y-\-c.,  as  the  gen- 
eral transformation  in  the  Gnfig,,  (cf.  p.  31)  we  have  TST"^ :  A-'=x-f-rt],  J''=)'+^l- 
Since  TST'^^S  we  have  a^=ij  a2=o  and  every  transformation  in  the  Gn  is  of 
the  form  T^:  x'=x-^ay-\-b,  y'^cy-\-el.  If  r=/  T,  is  of  period  2  if  cf!=o  and 
of  period  4  if  ad^i.  If  c^i  or  r  and  b=acd ,  T,  is  of  period  ■>.  If  c=i  or  /- 
and  b  is  not  equal  to  acd,  T  is  of  period  0.  Hence  every  transformation 
in  G„  is  of  type  II,  III,  IV  or  V.  Also,  all  transformations  of  types  III  and  V 
have  for  center  the  point  4^(/,o,o)  which  is  the  center  of  S,  all  homologi(?s 
have  for  axis  some  line  other  than  /  through  4  and  for  center  some  point  other 
than  4  on  /,  and  all  transformations  of  type  II  have  4  as  the  point  of  intersection 


189455 


38  U.  G.   Mitchell:   Geometry  and 

of  the  two  invariant  lines.  Since  no  homology  has  /  for  axis,  there  are  but  two 
transformations  (S  and  the  identity)  in  G„  which  leave  /  point-wise  invariant. 
Gn  is,  therefore,  at  most  (2,  1)  isomorphic  with  some  group  in  one  dimension 
leaving  a  point  on  the  line  invariant  and  its  possible  orders  are  (cf.  p.  32)  4,  8, 
12  and  24.  That  G„  can  be  a  cyclic  G^  follows  from  the 
fact  that  every  system  of  transitivity  of  the  (j^  contains  but  two 
points.  Gn  can  not  be  a  G^  whose  transformations  are  all  of  period 
2,  since  if  E  be  one  of  the  elations  in  such  a  G^,  the  Go  and  the  G^  have  the  same 
systems  of  transitivity  on  the  axis  of  E.  If  G,i  be  of  order  8  and  contain  trans- 
formations of  period  2  only  it  must  contain  two  elations,  Ej  and  E.,,  having  the 
same  axis  /j,  since  such  a  Gg  must  contain  six  elations  and  there  are  but  four 
lines  other  than  /  through  4.  We  then  have  EiE2=E3  a  third  elation  having  l^ 
for  axis.  Since  E^,  E.,  and  E,  have  the  same  center  and  axis  they,  together  with 
the  identity,  form  a  group  G4(P).  The  other  three  elations  in  the  G^  would  be 
SEj,  SEo  and  SE, ;  but  since  each  of  these  elations  has  the  same  systems  of  tran- 
sitivity on  /i  as  S,  Gn  can  not  be  such  a  G^.  Accordingly,  if  Gn  be  of  order  8  it 
must  contain  a  cyclic  subgroup  G^  consisting  of  the  powers  of  a  transformation  U 
of  type  III  and  since  U  must  leave  /  invariant  we  have  IJ-^S.  Hence  U  is  of 
the  form  U :  x'=x-\-ay-\-b,  y'==y-\-a'  where  a  is  not  zero.  If  the  G^  contain  an 
elation  it  must  be  of  the  form  E:  x'=x-{-ayy-\-by,  y'=y  where  rt^  is 
not  zero.  If  a=a^  the  product  EU:  x^^x-\-b-\-bj^-\-i,  y^=y-\-fi-  is  a 
translation  different  from  S.  If  a  is  not  equal  to  «,  the  product 
KU :  x'=x-{-(a-\-aj^)y-\-b-]-b^-\-a^a'-,  y'=y-\-^i'  i-^  a  transformation  of  t\pe 
III  whose  square  must  be  identical  with  S.  If  (EU)-:  a-'=a-j- 
a^a--\-i,  y^=y  be  identical  with  S  we  must  have  a^a--\-I=I  or  a^a'^-\-i=^o;  but  i{ 
a^^a--\-l=l,  a^a-=0  which  is  impossible  since  neither  a  nor  <7j  can  be  zero, 
and  if  aia--\-i=o,  a^d-=i  which  is  impossible  since  a  and  a^  are  different  marks 
of  GF(2-).  Hence  if  Gn  be  of  order  8  it  can  contain  no  elations 
and  must  have  3  cyclic  subgroups  of  order  4.  Taking  \] ^\  x'=x-\-aiy-\-b^,  y'^= 
y-f-rtj-  as  anv  other  transformation  of  period  4  than  U  or  U'  in  the  Gg  we  have 
UU,:  x'=x-]-{a,+a,)y-\-aa;'^b+b„  y'=y^a--\-a,'  ?Lnd  U^U :  x'^x^{a^a,)y 
-\-d-a^-\-b-\-b^,  y'=y-\-ar-\-a^-.  If  a^a■^  the  product  UU,  is  a  translation  not 
in  the  G^.  Hence  a  must  be  different  from  a^  and  the  group,  if  existent,  is  not 
Abelian  and  must  be  of  the  type'^  U*=l,  Ui*=l,  U-=Ui-,  UUiU-^= 
Ui"^  Since  these  conditions  are  satisfied  by  any  two  transformations  of  the  form 
U  and  Ui,  where  a  is  different  from  «,  and  neither  a  nor  a^  is  zero,  there  are 
four  such  groups  Gg  having  the  given  G^  as  the  largest  self-conjugate  subgroup 
of  translations. 

If  Gn  be  of  order  12  it  must  be  simply  isomorphic  with  the  G,.  consisting  of  all 
transformations  leaving  in\ariant  a  point  on  the  line '(cf.  Ill,  p.  32).  This  is 
impossible  since  the  product  of  the  translation  in  Gn  and  any  homology  in 
Gn  is  of  period  G.  Hence  Gn  can  not  be  of  order  12.  If  Gn  be  of  order  24  it 
must,  by  Theorem  5,  contain  exactly  8  transformations  of  determinant  unity  and  IG 
transformations  of  determinant  /  or  r.  The  8  transformations  of  determinant 
unity  form  a  self-conjugate  subgroup  and  hence  the  G04,  if  existent,  must  be  the 
direct  product  of  this  G^  and  a  cyclic  G..  consisting  of  the  powers  of  a  homology 
H,  It  was  shown  above  that  a  G.,  whose  transformations  are  all  of  period  2  con- 
tains 3  elations  E,,  E/,  E/',  having  the  same  axis  /^  and  three  other  elations  Eo, 
E3,  E^,  having  for  axes  the  three  other  lines  /,.  ''3.  h'  respectively,  through  P.     H 

*Cf   BURNSIDE,   1.   c,   p.   88. 


COLUNEATION    GrOUI'S   OF    PG('2;2-).  39 

can  not  have  /,  for  axis  since  the  G,  and  the  G04  would  then  have  the  same  systems 
of  transitivity  on  /p  Also  H  can  not  have  /o,  /..  or  l^  for  axis,  since  if  E, 
(/=-^2,;j,4)  be  the  corresponding  elation,  Ei  and  H  are  not  commutative  and 
HEjH'^  would  be  an  elation  not  in  the  Gj^.  Hence  the  self-conjugate  Ci^  in 
the  G,4  can  not  have  all  its  transformations  of  period  2.  Accordingly,  the  G^  in 
the  G24  must  contain  3  cyclic  subgroups  of  order  4.  We  may  take  this  self-con- 
jugate Gg  to  be  the  one  consisting  of  all  transformations  of  the  form  x'=x-^ay-\-k, 
j/==j;-|-rt-  where  a  is  any  mark  of  the  GF(2'-)  and  k  is  any  mark  of  the  GF(2). 
The  three  transformations  in  this  G^  of  the  form  T:  x'^x-\-ay,  y'=y-\-d-  where 
a  .is  not  zero  are  of  period  4  and  no  two  belong  to  the  same  cyclic  G^.  The  G^..i. 
if  existent,  must  contain  a  homology  of  the  form  H:  x'^=x-\-by-\-bic  y'^iy-\-c 
and,  therefore,  every  product  TH :  x'=xAr{ia-\-b)y-\-{a-\-ib)c,  y'=iy-\-(r-\-c. 
where  b  and  c  are  some  two  chosen  marks  of  the  GF(2'-).  Since  TH  is  of  deter- 
minant t  it  is  of  type  H  or  IV  and,  hence,  (TH)  ':  x^=x-\-i{a-b-\-nc^i) ,  y'=y 
must  be  identity  or  the  translation  S:  x'=x-{-ij  y'=y  for  every  value  of  a  different 
from  zero  in  the  GF(2-).  But  there  exist  no  two  marks  b  and  c  in  the  GF(2-) 
which  make  i{d-b-\-ac-\-i)=m  where  in  is  in  the  GF(2)  for  every  value  of  a  dif 
ferent  from  zero  in  the  GF(2-).     Hence,  G„  can  not  be  of  order  24. 

Subgroups  having  a  G^{Q)  as  a  self-conjugate  subgroup.  Let  Gm  be  a  sub 
group  of  G2S80  having  a  G4(Q)  as  its  largest  self-conjugate  subgroup  of  transla- 
tions and  such  that  the  G4(Q)  has  no  sj'stem  of  transitivity  which  is  also  a  system 
of  transitivity  of  the  G,„.  Selecting  the  G^CQ)  as  the  group  of  all  translatibns 
of  the  form  S:  x'=x-\-a,  y^=y-\-b,  where  a  and  b  are  marks  of  the  GF(2),  and 
T  is  the  general  collineation  (cf.  p.  31)  in  the  Gosso  we  have  TST"^ : 
x'=x-\-aa-i^-\-bb^,  y'=y-\-aa2-^bb.,.  Hence  if  TST"^  belong  to  the  G4(Q)  aa^-^- 
hb^^=a'  and  aa.^-\-bb._,=y  where  a'  and  b'  are  in  the  GF(2).  Since  these  equa- 
tions must  hold  true  for  every  a^,  a-,,  bj.  b^  in  the  Gm  no  matter  what  marks  of  the 
GF(2)  a^  and  Z*'  may  be  it  follows  that  a^,  a^,  b^,  bo  are  in  the  GF(2)  and  every 
transformation  in  the  Gm  is  of  determinant  unity.  Consequently.  Gm  can  contain 
no  homology  and  is  at  most  (4,  1)  isomorphic  with  some  group  on  the  line  leaving 
invariant  a  pair  of  points.  The  possible  orders  of  Gm  are,  therefore,  (cf.  p.  32) 
12  and  24.  The  G4(Q)  leaves  each  point  on  /  {x.^=o)  invariant  and  permutes 
the  other  16  points  in  four  systems  of  transitivity  each  consisting  of  four  points  no 
three  of  which  are  collinear.  These  four  quadrangles  are  0,^^(0  16  20  18) 
Q3=(l  14  8  5)  ;  Q2=(2  13  6  15)  ;  Q^=(  9  10  12 ll)  (cf.  Table  of  alignment. 
p.  3).  Every  transformation  in  Gm  is  of  the  form  T:  x'=a^x-\-b^y-\-Ci,  y'= 
rt.x-j-Z'gj'-l-Ca  where  a^,  a.,,  b^,  bn  are  in  the  GF(2),  and  must,  therefore,  be  of  type 
V  (elation,  period  2),  type  I^  (period  3)  or  type  HI  (period  4).  From  the  forms 
of  T",  T",  T^  it  appears  that  every  elation  in  Gm  must  be  of  the  form 
E-^':  x'^y-\-kj  y'=x-\-k,  or  of  the  form  E^:  x'=x-\-y-\-c,  y'=y,  or  of  the  form 
E,:  Ar'=*',  >;'=x-|-3;-(-(^;  every  transformation  of  type  I3  must  be  of  the  form 
T, :  x'=^y-\-7n,  y'=x-\-y-\-n,  or  of  the  form  T, :  x'=x-\-y-\-r,  y'=x-^s;  every 
transformation  of  type  HI  must  be  of  the  form  U, :  x'=y-^k,  y'=x-\-l  where  / 
and  k  are  not  the  same  mark,  or  of  the  form  U^ :  x'^x-^y-^-c^,  j'=>'+f2»  or  t»f 
the  form  U, :  x'=x-\-d-^,  y'^x-^y'-\-d.^.  If  Gm  be  of  order  12  it  must  be  the 
direct  product  of  a  cyclic  G3(cyc.  I3)  and  the  G4(Q).  But  each  transformation 
of  period  3  in  Gm  must  leave  one  of  the  quadrangles  Qi(i=l,  2,  3,  4)  invariant 
and  permute  the  other  three  in  cyclic  order.  Hence  some  one  of  these  quadrangles 
would  be  a  system  of  transitivity  of  the  G.o  generated  by  any  G, (eye.  I..)  and  the 
G^{Q_)  and  Gm  can  not  be  or  order  12.     If  Gm   be  of   order  24  it   must  contain  a 


40  U.  G.    Mitchell:   Geomltry  axd 

transformation  T  of  period  3  leaving  invariant  a  quadrangle  Qa  (a=l,  3,  3  or  4). 
Every  transformation  of  the  form  Tj  or  To  is  seen  to  leave  invariant  the  two 
points  7^(/\/,c>)  and  19^( /,//>)  on  /,  and  since  every  translation  in  the  G4(Q) 
leaves  every  point  on  /  invariant  the  eight  products  obtained  by  multiplying  T  and 
T-  by  the  transformations  in  the  G4(Q)  are  eight  transformations  of  period  3 
leaving  Qa  invariant.  Moreover,  G04  can  not  contain  any  transformation  T^  of 
period  3  not  included  among  these  eight,  for  the  products  of  T  and  the  three 
other  transformations  leaving  Qa  invariant  and  making  the  same  permutation  of 
points  on  /  as  T  give  the  three  translations  in  the  G^CQ)  and  consequently  one 
of  the  products  TjT  or  TiT-  would  be  a  translation  not  in  the  G^IQ).  Hence, 
the  G4  must  contain,  besides  the  G4(Q),  8  transformations  of  type  I.,  leaving  Qj^ 
invariant  and  some  transformation  S,  of  period  2  or  4  transforming  Qa  into  one 
of  the  other  quatlrangles.  Let  Sj  be  of  period  2  or  4  interchanging  tlie  four  quad- 
rangles in  any  order  R,  =  (QaQb)  (QcQd)  where  a.  b.  c,  d  are  the  numbers 
1,  2,  3,  4  in  an  arbitrary  order.  Since  half  of  the  transformations  of  period  '3  in 
the  G04  must  make  the  transformation  R2=(Qa)  (QbQcQd)  on  the  four  quad- 
rangles Qi(/=1,  2,  3,  4)  the  G04  would  then  contain  a  transformation  of  period  3 
making  on  Qi(/^1,  2,  3,  4)  the  transformation  RoR,  =  (QaQhQc)  (Qd)  \\hich 
has  just  been  shown  to  be  impossible.  Hence  the  G.^  can  contain  no  transformation 
of  period  2  or  4  interchanging  the  Qj  (i=l,  2,  3,  4)  in  pairs.  Also,  G04  can  not 
contain  a  transformation  of  period  4  permuting  the  Qi  (/=!,  2,  3,  4)  in  cyclic 
order  for  its  square  would  be  a  transformation  of  period  2  interchanging  them  bi' 
pairs.  Hence,  the  G04  must  contain  a  transformation  S,  of  period  2  or  4  which 
transforms  the  Q,  (/=1,  2,  3,  4)  in  the  order  R,=  (QaQ.,)  ( Q.)  (Qd)  I  but 
since  R.R,;=(QaQbQcQd)  the  G04  would  then  contain  transformations  permut- 
ing the  Qi(/=1,  2,  3,  4)  in  cyclic  order  which  has  just  been  shown  to  be  impossible. 
Hence  G.^so  contains  no  subgroup  G^  having  a  G4(Q)  as  its  largest  self-conju- 
gate subgroup  of  translations  and  such  that  the  G4(Q)  has  no  system  of  transi- 
tivity which  is  also  a  system  of  transitivity  of  the  Gm. 

Subgroups  containing  a  self-conjugate  G^{F).  Let  Gu  be  a  subgroup  of  G_,^;;„ 
having  a  G4(P)  as  its  largest  self-conjugate  subgroup  of  translations  and  such  that 
the  G4(P)  has  no  system  of  transitivity  w^hich  is  also  a  system  of  transitivity  of 
the  Gu.  Selecting  the  G4(P)  as  the  group  of  all  transformations  of  the  form 
S:  x'=x-\-a,  .v'=.v.  and  T:  x'=a^x-\-b^y-\-c^,  y'=a.,-x-\-b.j'-]-c.,  as  the  general 
transformation  in  the  Go^so  we  have  TST"' :  x'^x-\-a^a,  y'=y-\-aa.^.  Hence  in 
order  that  TST''  may  belong  to  the  G,(P)  we  must  have  a.,=o 
and  if  we  also  have  «,  =  /  each  translation  in  the  G4(P)  is  self- 
conjugate.  Hence  every  transformation  in  Gu  is  of  the  form  T^ : 
Jf'=rt,x-|-^i>'+^i'  y'=b.,y~\-Cn.  The  G4(P)  consists  of  all  transla- 
tions having  /  for  axis  and  4^(/,r),o)  for  center.  The  G4(P)  has  four  systems 
of  transitivity,  /,=(0  1  16  14),  /,s(8  5  18  20),  /,=  (2  13  10  9).  /4=(15  16  i2  11^ 
and  in  each  system  the  four  points  are  collinear.  From  the  form  of  T,-  it  appears 
that  every  elation  in  Gi,  is  of  the  form  E:  A-'^x-)-/',.v-|-f, ,  >•'=)'  (where  ^,  is  not 
zero)  and.  consequently,  has  4  for  center  and  leaves  each  /i(i=l,  2,  3,  4)  invar- 
iant. Hence  there  is  no  Gu  containing  translations  and  elations  only.  From  the 
form  of  T/  it  appears  that  every  transformation  of  type  HI  in  Gu  is  of  the  form 
^ '•  •*'=-*'+^i.i'+^i'  .v'^>'+<"2'  where  neither  /»,  nor  fo  is  zero.  The  12  trans- 
formations for  which  r.=  /  make  the  interchange  ilj.,)(lj^)\  the  12  for  which 
'•2=/ make  the  interchange  (/, /.,)'( A,/,)  ;  and  the  12  for  which  fo=r  make  the  inter- 
change (/j/j)  (/;./.).    Since  the  Gu  can  not  leave  any  /i(/=l,  2,  3,  4)    invariant  it 


COLLINEATION    GrOUPS    OF    PG(2;2-),  41 

must  either  be  transitive  on  the  /;  or  interchange  them  by  pairs. 
Suppose  Gk  to  make  the  interchange  {/J.,){l.J^).  Such  a  G„  can 
not  contain  a  homology  having  /  for  axis,  for  the  homology  would  permute  three 
of  the  lines  /,  in  cylic  order.  Hence,  the  Gu  would  be  at  most  (4,  1) 
isomorphic  with  some  group  on  the  line  leaving  a  point  invariant  and  its  possible 
orders  would  be  (cf.  p.  32)  S,  12,  16,  24,  48.  Since  Gk-  is  transitive  on  S  points  its 
order  must  be  divisible  by  8  and  can  not  be  12.  If  Gu  be  of  order  S  and  make: 
the  interchange  {/J..){lj^)  it  must  contain  a  transformation  of  the  form  U^ : 
^'^^x-\-b,y-\-c^,  y'=y^i-  The  products  of  a  Uj  and  the  transformations  in  the 
GJP)  are  four  transformations  of  the  form  Uj  where  A,  is  fixed  and  c^  is  any 
mark  of  the  GF(2-).  These  4  transformations  of  type  III  and  the  transforma- 
tions in  the  G^(P)  form  a  group  G^  which  is  a  Gu.  Hence  there  are  three  such 
subgroups  Gs  (one  for  each  value  of  b^)  interchanging  the  /i(i=l,  2,  3,  4)  in  the 
order  (/,/o)(/J4)  and  similarly  3  subgroups  G,  for  each  of  the  orders  (/,/0 
(/.,/,)  and  {IJ^){IJ..).  No  Gu  can  contain  more  than  8  transformations  of  the 
form  U,,  since  the  product  of  Uj^:  x'=x-\-y-\~c^,  v'=J'-(-  V  / :  x'=x-\-iy-\-c^, 
/=v  +  /,  and  U/':  .v'=A-+rv+f,,  j''=v  +  /,  is  U,  U/  U/':  x'=.v4-f,+/,  3;'=- 
3-f-/,  which  is  a  translation  not  in  the  G4(P).  The  product  of  any  two  transfor- 
mations of  the  form  U,  and  U/  k.  UiU/=Ei:  x'=x-\-ry-\- 1,  y'=y  which  is  an 
elation  having  4  for  center.  The  products  of  E^  and  the  four  transformations  in  the 
G^tP)  give  all  elations  of  the  form  E/:  x'=x-\-i'y-\-c,  y'=y.  The  product  of 
any  two  elations  of  the  form  E/  is  a  translation  in  the  G^fP).  Also  all  prod- 
ucts of  transformations  of  the  form  E/  with  transformations  of  the  form  Ui  or 
U/  are  of  the  form  U/  or  U,  respectively.  Hence,  the  12  transformations  of 
the  forms  U,,  U/  and  E/  together  with  the  G4(P)  form  a  Group  Gj,,  which  is 
a  Gu.  Obviously  there  may  be  two  other  such  groups  Gjp,  making  the  interchange 
(/,  /^)(/..  l^)  and  similarly  3  groups  Gj,.,  making  the  interchange  (,/,  /.;)  (/o  /J 
and  3  groups  Gi,.,  making  the  interchange  (/,  li){L,  /y).  Also,  since  the  product 
of  an  elation  of  the  form  E:  x'^x-\-by-\-i\,  y'^y  and  a  transformation  of  type 
III  of  the  form  U:  .v'=a--j-/>'.v4-'"i.  y'=y-\-('-2  is  EU :  x'=-x-{-c-t-\-bc2-\-c,y^=y-l-c^, 
which  is  a  translation  not  in  the  G^CP),  there  is  no  other  type  of  group  of  order 
10  which  can  be  a  Gu  interchanging  the  /j  by  pairs. 

A  Gu  of  order  24  or  48  which  makes  the  interchange  (/jA.)  (A./^)  must  con- 
tain transformations  of  period  3.  Every  such  transformation  would  leave  each 
/j  invariant  and  hence  would  be  a  homology  H  having  4  for  center.  Such  a  Gk 
must  also  contain  some  transformation  T  making  the  interchange  {/,  /o)  (/•}  /4). 
But  T  can  not  be  of  type  II,  for  in  that  case  T'  would  be  a  translation  not  in  the 
G,(  P),  and  T  can  not  be  of  type  III,  for  in  that  case  HT  would  be  such  a  trans- 
formation of  type  II.  Hence  there  is  no  Gu  of  order  24  or  48  interchanging  the 
/,  by  pairs.. 

No  Gu  can  contain  a  homology  H  having  /  for  axis;  for,  if  P,  be  the  center  of 
H,  Gk  must  contain  some  transformation  T  transforming  Pj  to  some  point  P/ 
which  is  not  collinear  with  Pj  and  4.  Since  H  can  not  leave  P/  invariant  HT 
and  TH  transform  P,  to  different  positions  and.  hence,  H  and  T  are  not  commu- 
tative THT"^=--H,  is,  therefore,  a  homology  having  P,  for  center  and  one  of 
the  products  HH,  or  H-H,  is  a  translation  not  in  the  G^fP)  since  it  would  have 
for  center  the  point  where  the  line  P,P/  cut  /.  Accordingly  Gu  is  at  most  (4,  1) 
isomorphic  with  some  group  on  the  line  leaving  a  point  invariant  and  its  possible 
orders  are  8,  12,  16,  24  or  48.  If  Gu  be  transitive  on  the  /j  it  is  transitive  on  all 
points  not  on  /  and  its  order  must,  therefore,  be  divisible  by  16.  Consequently  such 
a    transitive     Gu     must     be     of     order     16     or     48.       If     such     a     Gu     be     of 


42  U.  G.    Mitchell:   Geometry  and 

order  16  it  must  contain  two  transformations  of  type  III.  U;: 
x'^x-\-a^y-\-b^,  y'=y-\-c^,  and  Uo".  x'^x-\-a.jy-j-b.,,  3''=J'+r..  where  r,  and  c.^  are 
neither  one  zero  and  are  distinct  from  each  other.  The  product  UjUo  is  of  t<he 
form  Us :  x'=x-\-a^y-\-b^,  y'^y-\~c^  where  a.^  is  different  from  a,  and  fi.,  and 
^3  is  different  from  c^  and  c.^.  Also  «,  must  be  distinct  from  a.,,  for  otherwise  U;; 
is  a  translation  not  in  the  G^CP).  All  products  U0U3  are  of  the  form  U,  and 
all  products  U,U.,  are  of  the  form  U..  Also,  the  square  of  each  Ui(/=1,  2,  3) 
is  a  translation  in  the  G^CP).  Hence  12  transformations,  -4  each  of  the  forms  U,, 
U^,  U;.,  such  tliat  a^,  a.^,  a.  are  no  two  the  same  mark  and  c-^,  c.,,  c.,_.  are  no  two 
the  same  mark,  form,  together  with  the  G^CP)  a  group  and  the  only  type  of  group 
G,,.  which  is  a  transitive  Gk  of  order  16.  Since  a^  and  f,  may  each  be  chosen  in  3 
different  ways  and  a.,  and  c,  may  each  then  be  chosen  in  2  different  ways  thcrt 
are  36  such  groups  G,,..  having  the  given  G.(P)  as  its  largest  self -con  jugate  sub- 
group of  translations. 

If  Gk  be  a  transitive  group  of  order  48  it  must  contain  a  transformation  T  of 
period  3.  If  T  be  a  homology  it  must  either  have  some  line  through  4  for  axis  or 
have  4  for  center.  That  T  can  not  be  a  homolog\%  having  /  for  axis  has  been 
shown  above.  That  T  can  not  be  a  homology  having  any  other  line  through  4  for 
axis  follows  from  the  fact  that  if  S  be  one  of  the  translations  in  the  G^(P)  TS  is 
of  type  II  and  (TS)'  is  a  translation  not  in  the  G^(P).  If  T  be  a  homology' 
having  4  for  center  it  leaves  each  /i(/=l,  2,  3,  4)  invariant.  Since 
the  G4^  must  contain  a  subgroup  G,,.,  which  can  have  no  transforma- 
tion other  than  identity  in  common  with  the  cyclic  G.,  generated  by  T,  the 
group  -)  Gjg,  G3  (is  the  G^^  and  leaves  each  /i(/=l,  2,  3,  4)  invariant  unless 
the  G,(;  contain  a  transformation  U  of  type  III  having  4  for  center  and  having  for 
axis  a  line  through  P,  some  point  on  /  different  from  P,  and  P.,.  Hence  the  G,,, 
would  contain  at  least  24  distinct  homologies  and  since  the  products  of  these  by  U 
give  24  distinct  transformations  of  type  II  the  G^^  \vould  contain  more  than  4(S 
transformations.     Hence  a  transitive  G^  of  order  4S  can  not  contain  a  homology. 

Since  T  can  not  be  a  homology  it  must  be  of  type  I.  having  for  vertices  of  its 
invariant  triangle  a  point  A  not  on  /  and  two  points,  4  and  some  other  point  P,. 
other  than  4  on  /.  The  cyclic  G.,  generated  by  T  together  with  the  G^tP)  gen- 
erates a  Gjo  consisting  of  all  transformations  of  determinant  unity  leaving  invari- 
ant the  points  4  and  Pj  and  the  line  4  joining  A  to  4.  But  the  G^,  must  contain 
a  subgroup  G,c  having  no  transformation  other  than  identity  in  common  with  the 
cyclic  G.,  generated  by  T.  Hence  the  G,,.,  and  the  cyclic  G..  would  generate  a  G^^ 
leaving  4  invariant  unless  the  G,,,  contain  a  transformation  U  of  type  III  inter- 
changing /a  with  some  other  line  /b  through  4.  If  the  other  two  lines  ilhrough  4 
be  designated  as  /c  and  /<.,  U  makes  the  transformation  \j  =  {IJ\,)  (hU)  t^- 
these  lines  and  T  makes  the  transformation  T=(/a)  (/b^^d)-  Hence  TU== 
{hhlb){U)  is  of  type  lo  leaving/,,  invariant  and  UT=(/a/b/d)  (/c)  is  of  type 
I.,  leaving  4  invariant.  Also,  the  product  of  TU  and  UT  is  a  transformation  of 
type  I.,  leaving  the  line  /a  invariant.  Each  of  these  transformations  of  type  I„  generates 
a  cyclic  G,,  which,  taken  with  the  G^CP)  generates  a  G,.,  containing  S  transfor 
mations  of  type  I...  These  32  transformations  of  type  I.,  are  such  that  each  point 
not  on  /  is  an  invariant  point  of  two  of  them  and  each  point  on  /  ( other  than  4 ) 
:s  an  invariant  point  of  8  of  them.  Hence  the  self-conjugate  G,,;  can  not  contain 
an  elation  E ;  for,  if  T'  be  a  transformation  of  type  I.,  having  an  invariant  point 
on  the  axis  of  E,  ET'  would  be  a  transformation  of  type  I3  not  among  the  32 
above  named.     The  G,,,  must,  therefore,  consist  of  12  transformations  of  t\pe  III 


COLLINEATION    GrOUI'S    OF    PG(2,2-).  43 

and  the  G4(P).  Taking  the  Gj,.  as  all  transformations  of  the  form  U  :  x'=x-\-ay-{-b, 
y'=yJf-a  and  the  cylic  G..  generated  by  a  transformation  of  type  I3  as  all  transfor- 
mations of  the  form  T:  x'^/nx,  y'=n  ,  where  mn^i  and  neither  rn  nor  n  is 
unity,  the  products  UT:  x'=mx-\-any-\-b,  y'=ny-\-a  and  TU :  x'=mx-\-amy-\- 
mh,  y'=ny-\-na  are  32  transformations  of  type  L  as  above  described.  Also,  it  is 
readily  verified  that  the  product  of  any  two  transformations  of  the  form  U,  UT 
and  TU  is  of  the  form  U,  UT  or  TU.  Hence  they  form  a  group  G48  which  is  a 
transitive  Gu. 

Subgroups  containing  a  self-conjugate  G^.  There  remains  to  be  determined 
every  subgroup  Gt  of  Gogso  which  leaves  fixed  no  point  not  on  I,  has  a  Gg  as  its 
largest  self-conjugate  subgroup  of  translations  and  has  no  system  of  transitivity  which 
is  also  a  system  of  transitivity  of  the  G^.  A  Gg  of  translations  consists  of  a  G4(P) 
(all  translations  of  which  have  for  center  the  same  point  P  on  /)  and  4  other 
translations  each  having  a  different  center  from  any  other  in  the  Gg.  The  Gg 
leaves  invariant  besides  /  and  P  a  pair  of  lines  through  P.  Hence  the  Gg  has  two 
systems  of  transitivity  of  S  points  each  and  a  Gt  must  be  transitive  on  the  16  points 
not  on  /.  No  Gt  can  contain  a  homology  H  having  /  for  axis;  for,  if  T  be  a  trans- 
lation in  the  Gg  having  a  point  P'  different  from  P  for  center  H  and  T  are  not 
commutative  (since  HT  and  TH  transform  the  center  of  H  to  different  positions) 
and  HTH"'  is  a  translation  having  P'  for  center  and  not  in  the  Gg.  Hence,  a  G. 
contains  no  transformation  leaving  /  pointwise  invariant  except  the  translations  in 
the  Gg  and  is  at  most  (8,  1)  isomorphic  with  some  group  leaving  invariant  a  point 
on  the  line.  Its  possible  orders  are,  therefore,  (cf.  p.  32)  16,  32,  4S  and  96.  Tak- 
ing P  as  the  point  4=(/,6',o)  the  G4(P)  in  the  Gg  becomes  the  four  translations 
of  the  form  S :  x'=x-\-a,y'=y'  and  the  four  other  translations  in  the  Gg  may  be 
taken  as  the  four  translations  of  the  form  S^:  x^==x-\-a,  y'=y^i.  Since  a  Gt  can 
contain  no  translation  not  in  the  Gg  a  translation  of  the  form  S  must  be  trans- 
formed into  a  translation  of  the  form  S,  and  a  transformation  of  the  form  S,  must 
be  transformed  into  a  translation  of  the  form  Sj  by  every  transformation  in  Gt.  It 
has  already  appeared  above  that  the  first  of  these  conditions  requires  that  every 
transformation  in  a  Gt  shall  be  of  the  form  Tj :  x'=a^xA^b^y-\-c^,  y'=b.,y^(.., 
Transforming  S^  through  Tj  gives  TS,T/^ :  x'=-x-\-aa^,  v'^;'-|-^2  ^"d  hence  a^^^l, 
b,^=l  and  every  transformation  in  Gt  is  of  the  form  To :  A-'=A"-f-^i.V  +  (i,  >''=jH-c..- 
If  b^-=o  or  if  Co=o  To  is  of  period  2.  Hence,  every  elation  in  a  Gt  is  of  the  form 
E:  x'^x-\-bj-\^c^,y'^yj  where  b^  is  not  zero.  If  neither  b^  nor  c,  is  zero  T^  is 
of  period  4.  Accordingly  a  Gt  can  contain  only  transformations  of  period  2  or  -1 
and  must  be  of  order  16  or  32. 

The  two  systems  of  transitivity  of  the  Gg  are  left  invariant  by  every  elation  of 
the  form  E  and  hence  a  Gt  must  contain  a  transformation  U  of  type  III.  Under 
the  Gg  of  translations  the  pair  of  lines  y=o,  y=i  is  one  system  of  transitivity  and  the 
pair  of  lines  y=i,  y=i'-  (equations  in  non-homogeneous  coordinates)  is  the  other 
system.  Hence  U  must  be  of  the  form  Uj  :  x'=x-]-ay-{-b,  y'=y-\-i  or  of  the  form 
Uo:  x'=x-\-ny^b,  y'=y-\-i'.  But  the  product  of  a  translation  of  the  form  Si 
and  a  transformation  of  the  form  U^  or  Uo  is  of  the  form  Uo  or  U,,  respectively, 
and  hence  every  Gt  must  contain  transformations  of  both  forms.  A  Gt  of  order 
16  must,  therefore,  contain,  besides  the  Gg  of  translations,  4  transformations  of  the 
form  Uj  and  4  of  the  form  U,  where  a  has  the  same  value  for  all  of  these  trans- 
formations of  type  III.  Since  every  product  U,Uo  and  UoUj  is  a  translation  in 
the  G,  these  16  transformations  form  a  Group  G^^  which  is  a  Gt.  Since  there  are 
3  choices  for  a  there  are  3  such  groups  (j,,;  liaving  the  given  G^  as  the  largest  sub- 


44 


U.  G.    Mitchell:   Geometry  and 


2;roup  of  translations.  Also,  it  appears  from  the  above  that  every  Gi  must  contain 
at  least  one  such  Gj...  as  a  subgroup.  This  Gj,,  will  be  taken  as  the  subgroup  con- 
sisting of  the  S  translations  in  the  G^,.  4  transformations  of  the  form  U/:  Ar'=.v-i- 
y-\-c,  .v'^.vH-/  and  the  four  transformations  of  the  form  U^':  x'=x-\-y-\-c, 
;'==y-|-r.     For  convenience  this  G,,..  will  be  referred  to  as  the  group  G'. 

A  Gi  of  order  32  must  contain  some  transformation  T  of  period  2  or  4  not  in 
the  subgroup  of  order  1(3  taken  as  G'.  If  T  be  of  period  2  it  must  be  of  the  form 
E:  x'=x-\-ay-j-b.  y'=y.  The  products  U/E:  x'=x-\r{(i-\-i)y-\-b-\-c, 
y'^y-{-i,  and  U2'E:  x'^x-\-{a-^i)y-{-b-\-Cj  y'=j'+r  are  translations  not  in  thr; 
G^  of  translations  if  «=/.  If  (i=i,  the  G.o  must  contain  4  transformations  of 
type  III  of  the  form  U/':  x'=x-\-ry-\-/n,  y'=y-\-i,  and  4  of  the  form 
Uo":  x'=x-i-i-\-\-rn,  y'=y-\-r.  Also,  the  products  of  E  and  the  translations  of 
the  form  S,  introduce  4  transformations  of  the  form  U^ :  x'=x-\-iy-\-b,  y'^y-\-T. 
Thus  are  determined,  besides  the  S  translations  in  the  G^,  4  transformations  of  each 
of  the  forms  E.  U/,  U,"  and  U...  These  are  all  of  the  form  U:  x'=x-\-ay-\-l/, 
Y'=y-\-c  where  c=i  or  /'■'  if  a=i  or  /-  and  <=(>  or  /  if  a=o  or  /.  Taking  U. : 
x'=x-\-ajy-\-bi,  y'=y-\-Ci  as  a  second  transformation  of  the  form  U  the  product 
is  UUi :  x'=x-\-(a.^-\-a)y-{-(ic^-\-h^-}-b,  y'=y-\-c-\-c^.  If  a=n  or  a^^O  or  if 
a^a,  it  may  be  verified  by  inspection  that  this  product  is  one  of  the  given  forms. 
For  the  other  possibilities  the  following  table  gives  the  results: 


<-/ 

li. 

r 

(■; 

til  suit i?ig  forms. 

/ 

i 

r  or  / 

o  or   / 

U/'  or  U," 

/ 

r 

r  or  / 

i-  or  / 

E  or  U3 

i 

I 

o  or   / 

r  or  / 

U/'  or  U.," 

i 

t' 

0  or   / 

i  or  ;-■ 

U/  or  U./ 

I' 

I 

/  or  /- 

/  or  /- 

E  or  U3 

r 

1 

7  or  /- 

"  or    7 

V  '  or  V  : 

Hence  these  •■>2  transformations  form  a  group  G...,  which  is  a  Gf 
If  a  in  E  had  been  chosen  as  r,  a  G...,  of  the  same  type  would  have  been  dete" 
mined.     Hence  there  are  two  groups  G.o  having  the  G'  as  a  subgroup. 


SUMMARY. 

1.  In  the  finite  projecti\e  plane  PG(2,2")  the  diagonal  points  of  a  complete 
quadrangle  are  collinear. 

'-.  If  an  outside  point  of  a  conic  be  defined  as  an\  point  of  intersection  of  tan- 
gents to  the  conic,  every  conic  in  the  PG(2, 2°)  has  but  one  outside  point  and  all  tan- 
gents to  the  conic  concur  at  that  point.  Through  every  point  other  than  the  outside 
point  there  passes  one  and  but  one  tangent  to  the  conic  and  every  line  through  the  out- 
side point  of  a  conic  is  a  tangent  to  the  conic. 

3.  In  the  PG(2/2")  six  and  but  six  points  can  be  chosen  such  tliat  no  three  of 
the  set  are  collinear. 

4.  All  of  the  types  of  projective  collineations  of  the  ordinary  projective  plane 
are  present  in  the  PG(2,2")  and  the  number  of  such  collineations  in  the  PG(2,2-) 
is  (10480. 


CoLLiNEATioN  Groups  OF  PG(2;J'),  45 

o.  Every  subgroup  of  the  group  0^0430  of  ^11  projective  collineatlons  in  the 
PG(2,2-;,  except  a  self-conjugate  Go„ioo  leaves  invariant  a  real  figure  [real  with- 
in the  PG(2,2-)j   or  an  imaginary  triangle. 

(i.     There  are  H  kinds  of  groups  leaving  invariant   an   imaginary   triangle  and 
their  list  is  given  in  Theorem  11. 

7.  All  configurations  in  the  PG(2,2-')  and  the  groups  characterizing  them  are 
determined.  These  groups  include  the  finite  groups  of  the  ordinary  projective 
plane.  Consequently,  the  simple  Go,,,,,  the  Hessian  Go,,,  and  the  simple  Gip,  are 
all  subgroups  of  the  G,.,„4so  ^"d  within  the  PG(2,2-)  the  geometric  invariant  of 
each-  is  a  real  configuration. 

S  The  subgroups  of  the  G.^^^.,^  which  leaves  a  line  invariant  are  chiefly  (1,0 
or  (-^l)  isomorphic  with  groups  on  the  line,  but  certain  groups  of  higher  isomor- 
phism arc  present  and  are  determined. 


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